Sample Questions for the Engineering Module Subtest “Formalising Technical Interrelationships“ In the subtest "Formalising Technical Interrelationships," you are to transfer technical or scientific facts described verbally into a formulaic presentation and to interrelate the arising parameters to each other. This test measures your ability to formalise, your deductive and combinatory powers and your ability to use basic mathematical tools of the trade. Deeper knowledge of mathematics and physics is not required to solve the problems – formulae and laws are given but must be used and interrelated correctly. Instructions Working time: 60 minutes In the following items, the relationships between various technical quantities will be described in a text or a sketch. Your task is to determine the formal relationship between the given quantities. Aids: – Circumference of a circle: = U r π 2 = π D – Area of a circle: A = = 2 r π π 4 2 D – Circle: degrees: 360° and arc: 2 π – Average speed: Distance divided by time – Rotational frequency: Number of revolutions per time unit (e.g. 10 revolutions per second or n = 10 s –1 ) – Pressure: Force divided by area – Torque: Force multiplied by lever arm (only applies to right angles) – A lever is balanced when the clockwise torque and the counter-clockwise torque are equal. – Proportionality: – The quantities x and y (e.g. weight and volume) of a body are proportional to one another ( if their quotient is a constant. ) y ~ x – The quantities u and (e.g. volume and pressure of an ideal gas at a constant temperature) are inversely proportional to one another when their product is a constant w ) ~ u ( 1 w
In the subtest Formalising Technical Interrelationships you are to transfer technical or scientific facts described verbally into a formulaic presentation and to interrelate the arising parameters to each other
This test measures your ability to formalise your deductive and combinatory powers and your ability to use basic mathematical tools of the trade Deeper knowledge of mathematics and physics is not required to solve the problems ndash formulae and laws are given but must be used and interrelated correctly
Instructions Working time 60 minutes
In the following items the relationships between various technical quantities will be described in a text or a sketch Your task is to determine the formal relationship between the given quantities
Aids ndash Circumference of a circle = U rπ2 = π D
ndash Area of a circle A = = 2rπ π4
2D
ndash Circle degrees 360deg and arc 2π
ndash Average speed Distance divided by time
ndash Rotational frequency Number of revolutions per time unit (eg 10 revolutions per second or n = 10 sndash1)
ndash Pressure Force divided by area
ndash Torque Force multiplied by lever arm (only applies to right angles)
ndash A lever is balanced when the clockwise torque and the counter-clockwise torque are equal
ndash Proportionality
ndash The quantities x and y (eg weight and volume) of a body are proportional to one another ( if their quotient is a constant )
y~ xndash The quantities u and (eg volume and pressure of an ideal gas at a constant
temperature) are inversely proportional to one another when their product is a constant
w
)~ u( 1
w
Trigonometry
ϕϕ
ϕϕϕϕϕ
tan1cot
cossintan1cossin 22 ===+
ϕ 0deg 30deg 45deg 60deg 90deg 120deg 150
deg 180deg
sinϕ = cos (90degndash ϕ )
0 21
22
23
1 23
21
0
γCba
sinsinsin==
βα (Sinussatz)
γcos2222 abbac minus+= (Kosinussatz)
The illustrations are merely included as a visualization aid and are not true to scale
ϕtan
ϕsin
ϕcos ϕ
1
α β
γa b
c
Example 1 A gear mechanism consists of the gears A and B Gear A has ZA cogs Gear B has ZB cogs In the time it takes Gear A to complete n
B
A number of rotations Gear B completes nBB number of rotations Which of the following equations is correct
(A) AA
BB nZ
Zn =
(B) B
AAB Z
nZn =
(C) A
BAB n
ZZn =
(D) A
ABB Z
nZn =
Degree of difficulty low
Example 2 In a steel mill sheet steel is rolled onto cylinders at the end of the production process When empty the radius of one of these cylinders is r0 and the cylinder turns at a constant rotation speed n during the rolling process The thickness of the sheet steel is expressed as d Which equation expresses the change in a cylinderrsquos radius r in relation to the time t (in seconds)
r0
d(A) dtrr += 0
(B) ( ) tndrr += 0
(C) ndtrr += 0
(D) t
ndrr += 0
Degree of difficulty medium Example 3 The initial weight of a rocket is WI After the engines are started (t=0) fuel is expelled the amount of fuel is proportional to time When the fuel has been burned up at the point in time T the engines are turned off The weight of the rocket has decreased to WT Which of the following equations applies for the rocket weight W at the point in time t in the time interval 0lt= t lt= T
(A)
(B)
(C)
(D) Degree of difficulty high
Solutions
Subtest ldquoFormalising Technical Interrelationshipsldquo Example 1 To solve this problem an equation is to be derived from the introductory text and then transformed As described in the text the time required by Gear A to rotate exactly nA times is equal to the time it takes Gear B to rotate nB number of times The following products can therefore be equated Z
B
A nA = ZBB nB B
To solve this equation for nB both sides must be divided by ZB BB Therefore the solution is the equation shown under (B)
Example 2 To solve this problem it is necessary to find a formula with which the value of a constantly changing variable (the radius of the cylinder) can be determined at any given point in time
Since the cylinder moves at a constant rotation speed n ndash this speed being defined as number of rotations per unit of time ndash n has to be multiplied by the time t The result (nt) indicates how often the cylinder has turned at this point in time
With every rotation of the cylinder one layer of steel sheet is added Therefore if the product nt is multiplied by the sheet thickness d the increase of the cylinderrsquos radius after t seconds can be determined
In order to calculate the total radius the radius r0 of the empty cylinder at the beginning of the rolling process must be added to the result
Alternative C is the only equation which reflects all of these aspects and is therefore the correct answer
Example 3 The task presented by this test item is to find an equation which describes the change in the rockets weight over the course of time To this end let us consider the following figure (see below) At the time of take-off (t = 0) the weight is WI After take-off fuel is expelled and the rockets weight decreases It can be deduced from the text that the amount of fuel expelled is proportional to time In other words in the time interval between 0 and T the weight decreases linearly (WI - WT) The slope of the resulting straight line is thus (WI - WT)T and is preceded by a minus sign because the weight is decreasing This line intersects the vertical axis at the point WI
The correct equation is therefore
Accordingly if
Subtest ldquoVisualising Solidsldquo
In the subtest Visualising Solids you have to infer perspectives of a solid from one given view of the solid The test measures your spatial sense Instructions Two parts Working time 30 minutes Question type 1
To solve the following items you are to visualize the bodies three-dimensionally In each exercise the body is shown from two perspectives You are to identify the view of the same body from a third perspective Please mark the correct solution (A B C or D) on your answer sheet
The viewsperspectives are referred to as follows
Parallel projection of a cube
Further pointers ndash In the illustrations all visible edges are depicted as continuous (uninterrupted) lines
ndash If the illustration of a view from the side is not accompanied by an arrow rarr indicating which of the two side views is intended part of the task is to find that out
ndash If for example a side view is illustrated to the right of the view from the front or the view from above it does not necessarily mean that it is a view from the right side
View from above (VA)
View from the front (VF)
View from the side (VS)
VA
VSVS(not visible
here)
Example 1 Given The view of a solid from above and one side view of the same solid
View from above (VA) View from one side (VS)
uarr
(VF)
Wanted View from the front (VF) of the solid
(A) (B)
(C) (D)
Degree of difficulty low
Example 2 Given The view of a solid from above and the view from the front of the same solid
View from above (VA) View from the front (VF)
uarr (VS)
Wanted View (VS) from the side of the solid indicated by the arrow
(A) (B)
(C) (D)
Degree of difficulty medium
Example 3 Given The View of a solid from above and the view from the front of the some solid
View from above (VA) View from the front (VF)
uarr (VF)
Wanted View from the side (VS) of the solid
(A) (B)
(C)
(D)
Degree of difficulty high
Question type 2
The following items also test your ability to visualise three-dimensional figures Each item consists of two illustrations showing a transparent cube with one or two cables in its interior The first illustration (left) always shows the view from the front In the picture on the right the same cube is illustrated again Your task is to determine whether the picture on the right shows that cube from the right (r) left (l) from below (w) above (a) or behind (d)
(A) r (B) l (C) w (D) a (E) d
Here you see the cube from the front
Here you see the cube from
In the picture on the right you see the cube from above On your answer sheet you would mark the (D)
These items can be solved in one of the following two ways
Imagine that the cube had been placed on a glass table and that you could walk all the way around it Standing to the right or left of the table you look at the cube from the right or from the left If you go behind the table you look at the cube from behind If you come back to the front of the table and bend over it to look at the cube from above and if you imagine yourself lying down underneath the table feet first you see the view from below
Or you imagine that you could pick up the cube and turn it around in your hands If you looked at the cube from the front ie from the position shown in the left-hand illustration and then tipped it towards you by 90 degrees not changing your own position at all then you would see the view from above If you looked at the cube from the front and then turned it 90 degrees to the right you would see the view from the left If you turned it from the starting position 90 degrees to the left you would see it from the right And if you turned it 180 degrees to the right or left from the starting position you would see it from behind Finally if you tipped it backward you would see it from below
Example 1
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty low Example 2
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty medium
Example 3
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front
Here you see the cube from
Degree of difficulty high
Solutions Subtest rdquoVisualising Solidsldquo Question type 1 Example 1 Visualise this solid as a tree stump which has been cut off diagonally When you look at it from above (view from above) you see that a fairly large piece has been cut out of its left half Behind the cut-out section however a relatively large section of the stump has remained standing You can therefore rule out Option (A) immediately because this option shows nothing remaining behind the cut-out section except for the outer bark Option (B) can likewise not be the correct solution since here only a section of bark has been removed from the front The piece cut out of Option (D) has only one straight side According to the view from above however the cut-out section has to have three straight sides This is the case only with the section cut out of Option (C) Therefore (C) is the correct answer Example 2 One means of solving this item is to begin by looking at a detail which does not occur in all four answer options Look for example at the figure which looks like an upside-down L on the left edge of Option (B) and (D) Does this figure result from the view from above and the view from the front Yes it does because the upside-down L is the unobstructed view of the high surface at the figurersquos centre Accordingly you can already rule out Options (A) and (C) Options (B) und (D) differ in that in Figure (B) a step has been indicated over the rectangular figure to the right whereas (D) shows a straight edge all the way to the top In the view from above however you can see the protruding section which forms the step Therefore (B) is the correct answer here
Example 3 In the process of solving this item it is initially unclear whether the side view we are looking for is a view from the left or one from the right On the basis of the view from above it can be deduced that in the view from the left side (VSL) one of the two protruding beams at the upper end of the figure points towards the viewer the other towards the right In the view from the right side (VSR) only one of the two protruding beams can be seen it points towards the left The answers (B) and (D) are therefore options for the VSL and the answers (A) and (C) are options for the VSR Answer (B) is out of the question because there is no line indicating an edge at the transition from the figurersquos base to the vertical element or ldquopillarrdquo on top of it ndash On the basis of the view from above it can be deduced that the base of the figure is square Therefore any view from the side would have to indicate the upper edge of the base just as the view from the front does In the case of answer (D) there is a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam In the view from above however this line cannot be accounted for In the case of answer (A) on the other hand there is likewise a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam Here that line is correct because this is one of the two options for a VSR The left vertical edge of the pillar must therefore be visible in front of the protruding beam The other elements of answer (A) also correspond with the view of the three-dimensional figure from above and the view of it from the side and (A) is accordingly the correct solution to this task Answer (C) cannot be the correct solution because the transition between the upper end of the pillar and the upper protruding beam is shown as a single plane According to the view from above however the vertical edge of the uppermost end of the pillar would have to be visible here Question type 2 Example 1 In the case of this simple example you can immediately rule out the perspective from below and above From below as well as from above you would be looking through a kind of tube Therefore the perspective illustrated on the right can only be the view from the right the left or behind Now look at the bottom end of the cable In the left-hand picture it faces you In the right-hand picture it faces away from you ie points in exactly the opposite direction Therefore it is clear that the right-hand picture shows the view from behind
Example 2 Here the only view you can rule out immediately is the one from behind (Option E) If the view from the front shows one end of the cable leading toward the back of the cube at the top right the view from behind would show this cable end coming at you at the top left This is not the case in the right-hand picture If you tip the cube forward in your imagination you immediately see that the correct answer cannot be the view from above and turning the cube 180 degrees or 90 degrees to the right also does not lead to the desired perspective But if you imagine yourself standing on the right side of the cube you see that the end of the cable which is concealed in the left-hand illustration comes toward you on the right side of the cube in the right-hand illustration Therefore right (A) is the correct answer
Example 3 Here the figure on the right cannot be showing the view of the cable from the left (B) In the view from the left the section of the cable running horizontally in the view from the front would have to be visible in the middle of the right-hand edge Answer (E) is incorrect for the same reason In the view from behind the horizontal section of cable would have to be
visible in the background likewise running horizontally about halfway between bottom and top
In the view from above (D) this same section of cable would have to be seen leading from one side to the other along the bottom surface of the cube The figure on the right cannot be showing the view from the right (A) because the part of the cable touching the upper left-hand wall in that figure would have to be touching the upper right-hand wall in the view from the front which is not the case The only remaining option is the view from below (C) which only reveals itself to be correct however upon closer examination We might easily find ourselves looking for the end of the cable clearly seen at the bottom left in the view from the front ndash In the view from below it runs right into a curve in the cable and thus appears not to be an end at all On the other hand the end of the cable visible on the right-hand edge in the view from below is not visible in the view from the front because it is hidden behind a curving section of cable
In the subtest Analysing Technical Interrelationships you have to analyse and interpret diagrams charts or tables depicting technical laws or formulae
The test measures the ability to abstract from scientific and technical facts and to put abstract facts in concrete terms Knowledge of mathematics physics or technology is not needed background information will be provided if necessary
Instructions Working time 60 minutes
These items contain questions from various technical areas Your task is to visualize simple technical procedures and recognize technical interrelationships Unless otherwise indicated the axes (scales) of all diagrams are linearly subdivided In some of the items you must identify the ldquoqualitativelyrdquo correct diagram In other words your task is to decide which graph best represents the relationship between the variables shown Even the correct diagram will not necessarily drawn to scale
Example 1 Arrangements I and II each include a beam which is pivot-mounted (like a swing or see-saw) A hook has been mounted on the right end of the beam The ends of the beam are connected by a rope which is threaded through rolls
beam beam
bearing hook
rope rope
hookbearing
A weight is hung on the hook Which of the following statements isare then correct (The masses of the beam rope and hook can be neglected)
I In the case of Arrangement I the right end of the beam moves downward II In the case of Arrangement II the right end of the beam moves downward
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty low
Example 2 This diagram shows the wattage required by a lift motor in a period of 12 minutes (min) When the lift travels upward four times as much wattage (per minute) is required as when the lift travels downward When the lift stops at a floor twice as much wattage (per minute) is required as when the lift travels downward The running time between two consecutive floors is 30 seconds At the point in time t = 0 the lift is on the third floor
Which of the following statements isare correct
I Within the 12 minutes shown the lift travels up to the 6th floor II At the point in time t = 10 min the lift is on the 3rd floor
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty medium Example 3 The diagram shows two thermometers on which no temperature scales have yet been indicated They are both filled with the same liquid and the amount of liquid is also the same Their tubes are of the same length However the tube of the left-hand thermometer has a narrower diameter than that of the right-hand thermometer
We will assume that the markings for the two temperature scales are added correctly they begin at the same height on each tube and end at the same height Both thermometers are used only at temperatures for which they are suitable
Which of the two statements is (or are) therefore correct
I Rises in temperature can be measured less accurately with the left-hand thermometer than with the right-hand one II The right-hand thermometer covers a greater temperature range than the left-hand one
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty high
Solutions Subtest ldquoAnalysing Technical Interrelationshipsrdquo Example 1 If there is no rope the right end of the beam moves downward in both cases when a weight is suspended from the hook So the question is whether this movement is prevented by the rope When a weight is suspended from the hook in Arrangement I the rope slackens at the right end and at the left end The right end of the beam moves downward and Statement I is accordingly correct When a weight is suspended from the hook in Arrangement II traction (pulling power) is applied to the right end of the rope By way of the rope this traction is transferred to the left end of the beam Since both ends of the beam are pulled downward with the same force the beam does not move Statement II is false The solution to this item is therefore A
Example 2 Three different values for the wattage P are shown in the diagram 1 2 and 4 According to the text P is the lowest when the lift travels downward In this case therefore P = 1 When the lift stops at a floor P = 2 When the lift travels upward P = 4
On the basis of this information it is possible to interpret the movement of the lift At the point in time t = 0 the lift is on the 3rd floor and stops there for 1 minute Then it travels downward for 1 minute Since it takes 30 seconds to travel one floor it is then on the 1st floor Following a stop of 1 minute it travels upward 2 minutes (corresponding to 4 floors) At the point in time t = 5 it is therefore on the 5th floor There it stops for 15 minutes then travelling upward to the 6th floor ndash Statement I is therefore correct One minute later it travels 15 minutes (corresponding to 3 floors) downward and at the point in time t = 95 is therefore on the 3rd floor ndash Statement II is accordingly also correct
The solution to this item is therefore C
Example 3 If the temperature is increased by x degrees the liquid inside each thermometer expands by the same volume However this increase in liquid volume makes the liquid rise to a higher level in the tube of the thermometer on the left Since the cross-section of the tube in the left-hand thermometer is smaller than that of the right-hand one a defined temperature change generally leads here to a greater change in the liquid level than with the right-hand
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
The solution to this item is therefore B
Trigonometry
ϕϕ
ϕϕϕϕϕ
tan1cot
cossintan1cossin 22 ===+
ϕ 0deg 30deg 45deg 60deg 90deg 120deg 150
deg 180deg
sinϕ = cos (90degndash ϕ )
0 21
22
23
1 23
21
0
γCba
sinsinsin==
βα (Sinussatz)
γcos2222 abbac minus+= (Kosinussatz)
The illustrations are merely included as a visualization aid and are not true to scale
ϕtan
ϕsin
ϕcos ϕ
1
α β
γa b
c
Example 1 A gear mechanism consists of the gears A and B Gear A has ZA cogs Gear B has ZB cogs In the time it takes Gear A to complete n
B
A number of rotations Gear B completes nBB number of rotations Which of the following equations is correct
(A) AA
BB nZ
Zn =
(B) B
AAB Z
nZn =
(C) A
BAB n
ZZn =
(D) A
ABB Z
nZn =
Degree of difficulty low
Example 2 In a steel mill sheet steel is rolled onto cylinders at the end of the production process When empty the radius of one of these cylinders is r0 and the cylinder turns at a constant rotation speed n during the rolling process The thickness of the sheet steel is expressed as d Which equation expresses the change in a cylinderrsquos radius r in relation to the time t (in seconds)
r0
d(A) dtrr += 0
(B) ( ) tndrr += 0
(C) ndtrr += 0
(D) t
ndrr += 0
Degree of difficulty medium Example 3 The initial weight of a rocket is WI After the engines are started (t=0) fuel is expelled the amount of fuel is proportional to time When the fuel has been burned up at the point in time T the engines are turned off The weight of the rocket has decreased to WT Which of the following equations applies for the rocket weight W at the point in time t in the time interval 0lt= t lt= T
(A)
(B)
(C)
(D) Degree of difficulty high
Solutions
Subtest ldquoFormalising Technical Interrelationshipsldquo Example 1 To solve this problem an equation is to be derived from the introductory text and then transformed As described in the text the time required by Gear A to rotate exactly nA times is equal to the time it takes Gear B to rotate nB number of times The following products can therefore be equated Z
B
A nA = ZBB nB B
To solve this equation for nB both sides must be divided by ZB BB Therefore the solution is the equation shown under (B)
Example 2 To solve this problem it is necessary to find a formula with which the value of a constantly changing variable (the radius of the cylinder) can be determined at any given point in time
Since the cylinder moves at a constant rotation speed n ndash this speed being defined as number of rotations per unit of time ndash n has to be multiplied by the time t The result (nt) indicates how often the cylinder has turned at this point in time
With every rotation of the cylinder one layer of steel sheet is added Therefore if the product nt is multiplied by the sheet thickness d the increase of the cylinderrsquos radius after t seconds can be determined
In order to calculate the total radius the radius r0 of the empty cylinder at the beginning of the rolling process must be added to the result
Alternative C is the only equation which reflects all of these aspects and is therefore the correct answer
Example 3 The task presented by this test item is to find an equation which describes the change in the rockets weight over the course of time To this end let us consider the following figure (see below) At the time of take-off (t = 0) the weight is WI After take-off fuel is expelled and the rockets weight decreases It can be deduced from the text that the amount of fuel expelled is proportional to time In other words in the time interval between 0 and T the weight decreases linearly (WI - WT) The slope of the resulting straight line is thus (WI - WT)T and is preceded by a minus sign because the weight is decreasing This line intersects the vertical axis at the point WI
The correct equation is therefore
Accordingly if
Subtest ldquoVisualising Solidsldquo
In the subtest Visualising Solids you have to infer perspectives of a solid from one given view of the solid The test measures your spatial sense Instructions Two parts Working time 30 minutes Question type 1
To solve the following items you are to visualize the bodies three-dimensionally In each exercise the body is shown from two perspectives You are to identify the view of the same body from a third perspective Please mark the correct solution (A B C or D) on your answer sheet
The viewsperspectives are referred to as follows
Parallel projection of a cube
Further pointers ndash In the illustrations all visible edges are depicted as continuous (uninterrupted) lines
ndash If the illustration of a view from the side is not accompanied by an arrow rarr indicating which of the two side views is intended part of the task is to find that out
ndash If for example a side view is illustrated to the right of the view from the front or the view from above it does not necessarily mean that it is a view from the right side
View from above (VA)
View from the front (VF)
View from the side (VS)
VA
VSVS(not visible
here)
Example 1 Given The view of a solid from above and one side view of the same solid
View from above (VA) View from one side (VS)
uarr
(VF)
Wanted View from the front (VF) of the solid
(A) (B)
(C) (D)
Degree of difficulty low
Example 2 Given The view of a solid from above and the view from the front of the same solid
View from above (VA) View from the front (VF)
uarr (VS)
Wanted View (VS) from the side of the solid indicated by the arrow
(A) (B)
(C) (D)
Degree of difficulty medium
Example 3 Given The View of a solid from above and the view from the front of the some solid
View from above (VA) View from the front (VF)
uarr (VF)
Wanted View from the side (VS) of the solid
(A) (B)
(C)
(D)
Degree of difficulty high
Question type 2
The following items also test your ability to visualise three-dimensional figures Each item consists of two illustrations showing a transparent cube with one or two cables in its interior The first illustration (left) always shows the view from the front In the picture on the right the same cube is illustrated again Your task is to determine whether the picture on the right shows that cube from the right (r) left (l) from below (w) above (a) or behind (d)
(A) r (B) l (C) w (D) a (E) d
Here you see the cube from the front
Here you see the cube from
In the picture on the right you see the cube from above On your answer sheet you would mark the (D)
These items can be solved in one of the following two ways
Imagine that the cube had been placed on a glass table and that you could walk all the way around it Standing to the right or left of the table you look at the cube from the right or from the left If you go behind the table you look at the cube from behind If you come back to the front of the table and bend over it to look at the cube from above and if you imagine yourself lying down underneath the table feet first you see the view from below
Or you imagine that you could pick up the cube and turn it around in your hands If you looked at the cube from the front ie from the position shown in the left-hand illustration and then tipped it towards you by 90 degrees not changing your own position at all then you would see the view from above If you looked at the cube from the front and then turned it 90 degrees to the right you would see the view from the left If you turned it from the starting position 90 degrees to the left you would see it from the right And if you turned it 180 degrees to the right or left from the starting position you would see it from behind Finally if you tipped it backward you would see it from below
Example 1
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty low Example 2
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty medium
Example 3
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front
Here you see the cube from
Degree of difficulty high
Solutions Subtest rdquoVisualising Solidsldquo Question type 1 Example 1 Visualise this solid as a tree stump which has been cut off diagonally When you look at it from above (view from above) you see that a fairly large piece has been cut out of its left half Behind the cut-out section however a relatively large section of the stump has remained standing You can therefore rule out Option (A) immediately because this option shows nothing remaining behind the cut-out section except for the outer bark Option (B) can likewise not be the correct solution since here only a section of bark has been removed from the front The piece cut out of Option (D) has only one straight side According to the view from above however the cut-out section has to have three straight sides This is the case only with the section cut out of Option (C) Therefore (C) is the correct answer Example 2 One means of solving this item is to begin by looking at a detail which does not occur in all four answer options Look for example at the figure which looks like an upside-down L on the left edge of Option (B) and (D) Does this figure result from the view from above and the view from the front Yes it does because the upside-down L is the unobstructed view of the high surface at the figurersquos centre Accordingly you can already rule out Options (A) and (C) Options (B) und (D) differ in that in Figure (B) a step has been indicated over the rectangular figure to the right whereas (D) shows a straight edge all the way to the top In the view from above however you can see the protruding section which forms the step Therefore (B) is the correct answer here
Example 3 In the process of solving this item it is initially unclear whether the side view we are looking for is a view from the left or one from the right On the basis of the view from above it can be deduced that in the view from the left side (VSL) one of the two protruding beams at the upper end of the figure points towards the viewer the other towards the right In the view from the right side (VSR) only one of the two protruding beams can be seen it points towards the left The answers (B) and (D) are therefore options for the VSL and the answers (A) and (C) are options for the VSR Answer (B) is out of the question because there is no line indicating an edge at the transition from the figurersquos base to the vertical element or ldquopillarrdquo on top of it ndash On the basis of the view from above it can be deduced that the base of the figure is square Therefore any view from the side would have to indicate the upper edge of the base just as the view from the front does In the case of answer (D) there is a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam In the view from above however this line cannot be accounted for In the case of answer (A) on the other hand there is likewise a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam Here that line is correct because this is one of the two options for a VSR The left vertical edge of the pillar must therefore be visible in front of the protruding beam The other elements of answer (A) also correspond with the view of the three-dimensional figure from above and the view of it from the side and (A) is accordingly the correct solution to this task Answer (C) cannot be the correct solution because the transition between the upper end of the pillar and the upper protruding beam is shown as a single plane According to the view from above however the vertical edge of the uppermost end of the pillar would have to be visible here Question type 2 Example 1 In the case of this simple example you can immediately rule out the perspective from below and above From below as well as from above you would be looking through a kind of tube Therefore the perspective illustrated on the right can only be the view from the right the left or behind Now look at the bottom end of the cable In the left-hand picture it faces you In the right-hand picture it faces away from you ie points in exactly the opposite direction Therefore it is clear that the right-hand picture shows the view from behind
Example 2 Here the only view you can rule out immediately is the one from behind (Option E) If the view from the front shows one end of the cable leading toward the back of the cube at the top right the view from behind would show this cable end coming at you at the top left This is not the case in the right-hand picture If you tip the cube forward in your imagination you immediately see that the correct answer cannot be the view from above and turning the cube 180 degrees or 90 degrees to the right also does not lead to the desired perspective But if you imagine yourself standing on the right side of the cube you see that the end of the cable which is concealed in the left-hand illustration comes toward you on the right side of the cube in the right-hand illustration Therefore right (A) is the correct answer
Example 3 Here the figure on the right cannot be showing the view of the cable from the left (B) In the view from the left the section of the cable running horizontally in the view from the front would have to be visible in the middle of the right-hand edge Answer (E) is incorrect for the same reason In the view from behind the horizontal section of cable would have to be
visible in the background likewise running horizontally about halfway between bottom and top
In the view from above (D) this same section of cable would have to be seen leading from one side to the other along the bottom surface of the cube The figure on the right cannot be showing the view from the right (A) because the part of the cable touching the upper left-hand wall in that figure would have to be touching the upper right-hand wall in the view from the front which is not the case The only remaining option is the view from below (C) which only reveals itself to be correct however upon closer examination We might easily find ourselves looking for the end of the cable clearly seen at the bottom left in the view from the front ndash In the view from below it runs right into a curve in the cable and thus appears not to be an end at all On the other hand the end of the cable visible on the right-hand edge in the view from below is not visible in the view from the front because it is hidden behind a curving section of cable
In the subtest Analysing Technical Interrelationships you have to analyse and interpret diagrams charts or tables depicting technical laws or formulae
The test measures the ability to abstract from scientific and technical facts and to put abstract facts in concrete terms Knowledge of mathematics physics or technology is not needed background information will be provided if necessary
Instructions Working time 60 minutes
These items contain questions from various technical areas Your task is to visualize simple technical procedures and recognize technical interrelationships Unless otherwise indicated the axes (scales) of all diagrams are linearly subdivided In some of the items you must identify the ldquoqualitativelyrdquo correct diagram In other words your task is to decide which graph best represents the relationship between the variables shown Even the correct diagram will not necessarily drawn to scale
Example 1 Arrangements I and II each include a beam which is pivot-mounted (like a swing or see-saw) A hook has been mounted on the right end of the beam The ends of the beam are connected by a rope which is threaded through rolls
beam beam
bearing hook
rope rope
hookbearing
A weight is hung on the hook Which of the following statements isare then correct (The masses of the beam rope and hook can be neglected)
I In the case of Arrangement I the right end of the beam moves downward II In the case of Arrangement II the right end of the beam moves downward
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty low
Example 2 This diagram shows the wattage required by a lift motor in a period of 12 minutes (min) When the lift travels upward four times as much wattage (per minute) is required as when the lift travels downward When the lift stops at a floor twice as much wattage (per minute) is required as when the lift travels downward The running time between two consecutive floors is 30 seconds At the point in time t = 0 the lift is on the third floor
Which of the following statements isare correct
I Within the 12 minutes shown the lift travels up to the 6th floor II At the point in time t = 10 min the lift is on the 3rd floor
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty medium Example 3 The diagram shows two thermometers on which no temperature scales have yet been indicated They are both filled with the same liquid and the amount of liquid is also the same Their tubes are of the same length However the tube of the left-hand thermometer has a narrower diameter than that of the right-hand thermometer
We will assume that the markings for the two temperature scales are added correctly they begin at the same height on each tube and end at the same height Both thermometers are used only at temperatures for which they are suitable
Which of the two statements is (or are) therefore correct
I Rises in temperature can be measured less accurately with the left-hand thermometer than with the right-hand one II The right-hand thermometer covers a greater temperature range than the left-hand one
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty high
Solutions Subtest ldquoAnalysing Technical Interrelationshipsrdquo Example 1 If there is no rope the right end of the beam moves downward in both cases when a weight is suspended from the hook So the question is whether this movement is prevented by the rope When a weight is suspended from the hook in Arrangement I the rope slackens at the right end and at the left end The right end of the beam moves downward and Statement I is accordingly correct When a weight is suspended from the hook in Arrangement II traction (pulling power) is applied to the right end of the rope By way of the rope this traction is transferred to the left end of the beam Since both ends of the beam are pulled downward with the same force the beam does not move Statement II is false The solution to this item is therefore A
Example 2 Three different values for the wattage P are shown in the diagram 1 2 and 4 According to the text P is the lowest when the lift travels downward In this case therefore P = 1 When the lift stops at a floor P = 2 When the lift travels upward P = 4
On the basis of this information it is possible to interpret the movement of the lift At the point in time t = 0 the lift is on the 3rd floor and stops there for 1 minute Then it travels downward for 1 minute Since it takes 30 seconds to travel one floor it is then on the 1st floor Following a stop of 1 minute it travels upward 2 minutes (corresponding to 4 floors) At the point in time t = 5 it is therefore on the 5th floor There it stops for 15 minutes then travelling upward to the 6th floor ndash Statement I is therefore correct One minute later it travels 15 minutes (corresponding to 3 floors) downward and at the point in time t = 95 is therefore on the 3rd floor ndash Statement II is accordingly also correct
The solution to this item is therefore C
Example 3 If the temperature is increased by x degrees the liquid inside each thermometer expands by the same volume However this increase in liquid volume makes the liquid rise to a higher level in the tube of the thermometer on the left Since the cross-section of the tube in the left-hand thermometer is smaller than that of the right-hand one a defined temperature change generally leads here to a greater change in the liquid level than with the right-hand
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
The solution to this item is therefore B
Example 2 In a steel mill sheet steel is rolled onto cylinders at the end of the production process When empty the radius of one of these cylinders is r0 and the cylinder turns at a constant rotation speed n during the rolling process The thickness of the sheet steel is expressed as d Which equation expresses the change in a cylinderrsquos radius r in relation to the time t (in seconds)
r0
d(A) dtrr += 0
(B) ( ) tndrr += 0
(C) ndtrr += 0
(D) t
ndrr += 0
Degree of difficulty medium Example 3 The initial weight of a rocket is WI After the engines are started (t=0) fuel is expelled the amount of fuel is proportional to time When the fuel has been burned up at the point in time T the engines are turned off The weight of the rocket has decreased to WT Which of the following equations applies for the rocket weight W at the point in time t in the time interval 0lt= t lt= T
(A)
(B)
(C)
(D) Degree of difficulty high
Solutions
Subtest ldquoFormalising Technical Interrelationshipsldquo Example 1 To solve this problem an equation is to be derived from the introductory text and then transformed As described in the text the time required by Gear A to rotate exactly nA times is equal to the time it takes Gear B to rotate nB number of times The following products can therefore be equated Z
B
A nA = ZBB nB B
To solve this equation for nB both sides must be divided by ZB BB Therefore the solution is the equation shown under (B)
Example 2 To solve this problem it is necessary to find a formula with which the value of a constantly changing variable (the radius of the cylinder) can be determined at any given point in time
Since the cylinder moves at a constant rotation speed n ndash this speed being defined as number of rotations per unit of time ndash n has to be multiplied by the time t The result (nt) indicates how often the cylinder has turned at this point in time
With every rotation of the cylinder one layer of steel sheet is added Therefore if the product nt is multiplied by the sheet thickness d the increase of the cylinderrsquos radius after t seconds can be determined
In order to calculate the total radius the radius r0 of the empty cylinder at the beginning of the rolling process must be added to the result
Alternative C is the only equation which reflects all of these aspects and is therefore the correct answer
Example 3 The task presented by this test item is to find an equation which describes the change in the rockets weight over the course of time To this end let us consider the following figure (see below) At the time of take-off (t = 0) the weight is WI After take-off fuel is expelled and the rockets weight decreases It can be deduced from the text that the amount of fuel expelled is proportional to time In other words in the time interval between 0 and T the weight decreases linearly (WI - WT) The slope of the resulting straight line is thus (WI - WT)T and is preceded by a minus sign because the weight is decreasing This line intersects the vertical axis at the point WI
The correct equation is therefore
Accordingly if
Subtest ldquoVisualising Solidsldquo
In the subtest Visualising Solids you have to infer perspectives of a solid from one given view of the solid The test measures your spatial sense Instructions Two parts Working time 30 minutes Question type 1
To solve the following items you are to visualize the bodies three-dimensionally In each exercise the body is shown from two perspectives You are to identify the view of the same body from a third perspective Please mark the correct solution (A B C or D) on your answer sheet
The viewsperspectives are referred to as follows
Parallel projection of a cube
Further pointers ndash In the illustrations all visible edges are depicted as continuous (uninterrupted) lines
ndash If the illustration of a view from the side is not accompanied by an arrow rarr indicating which of the two side views is intended part of the task is to find that out
ndash If for example a side view is illustrated to the right of the view from the front or the view from above it does not necessarily mean that it is a view from the right side
View from above (VA)
View from the front (VF)
View from the side (VS)
VA
VSVS(not visible
here)
Example 1 Given The view of a solid from above and one side view of the same solid
View from above (VA) View from one side (VS)
uarr
(VF)
Wanted View from the front (VF) of the solid
(A) (B)
(C) (D)
Degree of difficulty low
Example 2 Given The view of a solid from above and the view from the front of the same solid
View from above (VA) View from the front (VF)
uarr (VS)
Wanted View (VS) from the side of the solid indicated by the arrow
(A) (B)
(C) (D)
Degree of difficulty medium
Example 3 Given The View of a solid from above and the view from the front of the some solid
View from above (VA) View from the front (VF)
uarr (VF)
Wanted View from the side (VS) of the solid
(A) (B)
(C)
(D)
Degree of difficulty high
Question type 2
The following items also test your ability to visualise three-dimensional figures Each item consists of two illustrations showing a transparent cube with one or two cables in its interior The first illustration (left) always shows the view from the front In the picture on the right the same cube is illustrated again Your task is to determine whether the picture on the right shows that cube from the right (r) left (l) from below (w) above (a) or behind (d)
(A) r (B) l (C) w (D) a (E) d
Here you see the cube from the front
Here you see the cube from
In the picture on the right you see the cube from above On your answer sheet you would mark the (D)
These items can be solved in one of the following two ways
Imagine that the cube had been placed on a glass table and that you could walk all the way around it Standing to the right or left of the table you look at the cube from the right or from the left If you go behind the table you look at the cube from behind If you come back to the front of the table and bend over it to look at the cube from above and if you imagine yourself lying down underneath the table feet first you see the view from below
Or you imagine that you could pick up the cube and turn it around in your hands If you looked at the cube from the front ie from the position shown in the left-hand illustration and then tipped it towards you by 90 degrees not changing your own position at all then you would see the view from above If you looked at the cube from the front and then turned it 90 degrees to the right you would see the view from the left If you turned it from the starting position 90 degrees to the left you would see it from the right And if you turned it 180 degrees to the right or left from the starting position you would see it from behind Finally if you tipped it backward you would see it from below
Example 1
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty low Example 2
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty medium
Example 3
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front
Here you see the cube from
Degree of difficulty high
Solutions Subtest rdquoVisualising Solidsldquo Question type 1 Example 1 Visualise this solid as a tree stump which has been cut off diagonally When you look at it from above (view from above) you see that a fairly large piece has been cut out of its left half Behind the cut-out section however a relatively large section of the stump has remained standing You can therefore rule out Option (A) immediately because this option shows nothing remaining behind the cut-out section except for the outer bark Option (B) can likewise not be the correct solution since here only a section of bark has been removed from the front The piece cut out of Option (D) has only one straight side According to the view from above however the cut-out section has to have three straight sides This is the case only with the section cut out of Option (C) Therefore (C) is the correct answer Example 2 One means of solving this item is to begin by looking at a detail which does not occur in all four answer options Look for example at the figure which looks like an upside-down L on the left edge of Option (B) and (D) Does this figure result from the view from above and the view from the front Yes it does because the upside-down L is the unobstructed view of the high surface at the figurersquos centre Accordingly you can already rule out Options (A) and (C) Options (B) und (D) differ in that in Figure (B) a step has been indicated over the rectangular figure to the right whereas (D) shows a straight edge all the way to the top In the view from above however you can see the protruding section which forms the step Therefore (B) is the correct answer here
Example 3 In the process of solving this item it is initially unclear whether the side view we are looking for is a view from the left or one from the right On the basis of the view from above it can be deduced that in the view from the left side (VSL) one of the two protruding beams at the upper end of the figure points towards the viewer the other towards the right In the view from the right side (VSR) only one of the two protruding beams can be seen it points towards the left The answers (B) and (D) are therefore options for the VSL and the answers (A) and (C) are options for the VSR Answer (B) is out of the question because there is no line indicating an edge at the transition from the figurersquos base to the vertical element or ldquopillarrdquo on top of it ndash On the basis of the view from above it can be deduced that the base of the figure is square Therefore any view from the side would have to indicate the upper edge of the base just as the view from the front does In the case of answer (D) there is a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam In the view from above however this line cannot be accounted for In the case of answer (A) on the other hand there is likewise a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam Here that line is correct because this is one of the two options for a VSR The left vertical edge of the pillar must therefore be visible in front of the protruding beam The other elements of answer (A) also correspond with the view of the three-dimensional figure from above and the view of it from the side and (A) is accordingly the correct solution to this task Answer (C) cannot be the correct solution because the transition between the upper end of the pillar and the upper protruding beam is shown as a single plane According to the view from above however the vertical edge of the uppermost end of the pillar would have to be visible here Question type 2 Example 1 In the case of this simple example you can immediately rule out the perspective from below and above From below as well as from above you would be looking through a kind of tube Therefore the perspective illustrated on the right can only be the view from the right the left or behind Now look at the bottom end of the cable In the left-hand picture it faces you In the right-hand picture it faces away from you ie points in exactly the opposite direction Therefore it is clear that the right-hand picture shows the view from behind
Example 2 Here the only view you can rule out immediately is the one from behind (Option E) If the view from the front shows one end of the cable leading toward the back of the cube at the top right the view from behind would show this cable end coming at you at the top left This is not the case in the right-hand picture If you tip the cube forward in your imagination you immediately see that the correct answer cannot be the view from above and turning the cube 180 degrees or 90 degrees to the right also does not lead to the desired perspective But if you imagine yourself standing on the right side of the cube you see that the end of the cable which is concealed in the left-hand illustration comes toward you on the right side of the cube in the right-hand illustration Therefore right (A) is the correct answer
Example 3 Here the figure on the right cannot be showing the view of the cable from the left (B) In the view from the left the section of the cable running horizontally in the view from the front would have to be visible in the middle of the right-hand edge Answer (E) is incorrect for the same reason In the view from behind the horizontal section of cable would have to be
visible in the background likewise running horizontally about halfway between bottom and top
In the view from above (D) this same section of cable would have to be seen leading from one side to the other along the bottom surface of the cube The figure on the right cannot be showing the view from the right (A) because the part of the cable touching the upper left-hand wall in that figure would have to be touching the upper right-hand wall in the view from the front which is not the case The only remaining option is the view from below (C) which only reveals itself to be correct however upon closer examination We might easily find ourselves looking for the end of the cable clearly seen at the bottom left in the view from the front ndash In the view from below it runs right into a curve in the cable and thus appears not to be an end at all On the other hand the end of the cable visible on the right-hand edge in the view from below is not visible in the view from the front because it is hidden behind a curving section of cable
In the subtest Analysing Technical Interrelationships you have to analyse and interpret diagrams charts or tables depicting technical laws or formulae
The test measures the ability to abstract from scientific and technical facts and to put abstract facts in concrete terms Knowledge of mathematics physics or technology is not needed background information will be provided if necessary
Instructions Working time 60 minutes
These items contain questions from various technical areas Your task is to visualize simple technical procedures and recognize technical interrelationships Unless otherwise indicated the axes (scales) of all diagrams are linearly subdivided In some of the items you must identify the ldquoqualitativelyrdquo correct diagram In other words your task is to decide which graph best represents the relationship between the variables shown Even the correct diagram will not necessarily drawn to scale
Example 1 Arrangements I and II each include a beam which is pivot-mounted (like a swing or see-saw) A hook has been mounted on the right end of the beam The ends of the beam are connected by a rope which is threaded through rolls
beam beam
bearing hook
rope rope
hookbearing
A weight is hung on the hook Which of the following statements isare then correct (The masses of the beam rope and hook can be neglected)
I In the case of Arrangement I the right end of the beam moves downward II In the case of Arrangement II the right end of the beam moves downward
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty low
Example 2 This diagram shows the wattage required by a lift motor in a period of 12 minutes (min) When the lift travels upward four times as much wattage (per minute) is required as when the lift travels downward When the lift stops at a floor twice as much wattage (per minute) is required as when the lift travels downward The running time between two consecutive floors is 30 seconds At the point in time t = 0 the lift is on the third floor
Which of the following statements isare correct
I Within the 12 minutes shown the lift travels up to the 6th floor II At the point in time t = 10 min the lift is on the 3rd floor
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty medium Example 3 The diagram shows two thermometers on which no temperature scales have yet been indicated They are both filled with the same liquid and the amount of liquid is also the same Their tubes are of the same length However the tube of the left-hand thermometer has a narrower diameter than that of the right-hand thermometer
We will assume that the markings for the two temperature scales are added correctly they begin at the same height on each tube and end at the same height Both thermometers are used only at temperatures for which they are suitable
Which of the two statements is (or are) therefore correct
I Rises in temperature can be measured less accurately with the left-hand thermometer than with the right-hand one II The right-hand thermometer covers a greater temperature range than the left-hand one
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty high
Solutions Subtest ldquoAnalysing Technical Interrelationshipsrdquo Example 1 If there is no rope the right end of the beam moves downward in both cases when a weight is suspended from the hook So the question is whether this movement is prevented by the rope When a weight is suspended from the hook in Arrangement I the rope slackens at the right end and at the left end The right end of the beam moves downward and Statement I is accordingly correct When a weight is suspended from the hook in Arrangement II traction (pulling power) is applied to the right end of the rope By way of the rope this traction is transferred to the left end of the beam Since both ends of the beam are pulled downward with the same force the beam does not move Statement II is false The solution to this item is therefore A
Example 2 Three different values for the wattage P are shown in the diagram 1 2 and 4 According to the text P is the lowest when the lift travels downward In this case therefore P = 1 When the lift stops at a floor P = 2 When the lift travels upward P = 4
On the basis of this information it is possible to interpret the movement of the lift At the point in time t = 0 the lift is on the 3rd floor and stops there for 1 minute Then it travels downward for 1 minute Since it takes 30 seconds to travel one floor it is then on the 1st floor Following a stop of 1 minute it travels upward 2 minutes (corresponding to 4 floors) At the point in time t = 5 it is therefore on the 5th floor There it stops for 15 minutes then travelling upward to the 6th floor ndash Statement I is therefore correct One minute later it travels 15 minutes (corresponding to 3 floors) downward and at the point in time t = 95 is therefore on the 3rd floor ndash Statement II is accordingly also correct
The solution to this item is therefore C
Example 3 If the temperature is increased by x degrees the liquid inside each thermometer expands by the same volume However this increase in liquid volume makes the liquid rise to a higher level in the tube of the thermometer on the left Since the cross-section of the tube in the left-hand thermometer is smaller than that of the right-hand one a defined temperature change generally leads here to a greater change in the liquid level than with the right-hand
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
The solution to this item is therefore B
Solutions
Subtest ldquoFormalising Technical Interrelationshipsldquo Example 1 To solve this problem an equation is to be derived from the introductory text and then transformed As described in the text the time required by Gear A to rotate exactly nA times is equal to the time it takes Gear B to rotate nB number of times The following products can therefore be equated Z
B
A nA = ZBB nB B
To solve this equation for nB both sides must be divided by ZB BB Therefore the solution is the equation shown under (B)
Example 2 To solve this problem it is necessary to find a formula with which the value of a constantly changing variable (the radius of the cylinder) can be determined at any given point in time
Since the cylinder moves at a constant rotation speed n ndash this speed being defined as number of rotations per unit of time ndash n has to be multiplied by the time t The result (nt) indicates how often the cylinder has turned at this point in time
With every rotation of the cylinder one layer of steel sheet is added Therefore if the product nt is multiplied by the sheet thickness d the increase of the cylinderrsquos radius after t seconds can be determined
In order to calculate the total radius the radius r0 of the empty cylinder at the beginning of the rolling process must be added to the result
Alternative C is the only equation which reflects all of these aspects and is therefore the correct answer
Example 3 The task presented by this test item is to find an equation which describes the change in the rockets weight over the course of time To this end let us consider the following figure (see below) At the time of take-off (t = 0) the weight is WI After take-off fuel is expelled and the rockets weight decreases It can be deduced from the text that the amount of fuel expelled is proportional to time In other words in the time interval between 0 and T the weight decreases linearly (WI - WT) The slope of the resulting straight line is thus (WI - WT)T and is preceded by a minus sign because the weight is decreasing This line intersects the vertical axis at the point WI
The correct equation is therefore
Accordingly if
Subtest ldquoVisualising Solidsldquo
In the subtest Visualising Solids you have to infer perspectives of a solid from one given view of the solid The test measures your spatial sense Instructions Two parts Working time 30 minutes Question type 1
To solve the following items you are to visualize the bodies three-dimensionally In each exercise the body is shown from two perspectives You are to identify the view of the same body from a third perspective Please mark the correct solution (A B C or D) on your answer sheet
The viewsperspectives are referred to as follows
Parallel projection of a cube
Further pointers ndash In the illustrations all visible edges are depicted as continuous (uninterrupted) lines
ndash If the illustration of a view from the side is not accompanied by an arrow rarr indicating which of the two side views is intended part of the task is to find that out
ndash If for example a side view is illustrated to the right of the view from the front or the view from above it does not necessarily mean that it is a view from the right side
View from above (VA)
View from the front (VF)
View from the side (VS)
VA
VSVS(not visible
here)
Example 1 Given The view of a solid from above and one side view of the same solid
View from above (VA) View from one side (VS)
uarr
(VF)
Wanted View from the front (VF) of the solid
(A) (B)
(C) (D)
Degree of difficulty low
Example 2 Given The view of a solid from above and the view from the front of the same solid
View from above (VA) View from the front (VF)
uarr (VS)
Wanted View (VS) from the side of the solid indicated by the arrow
(A) (B)
(C) (D)
Degree of difficulty medium
Example 3 Given The View of a solid from above and the view from the front of the some solid
View from above (VA) View from the front (VF)
uarr (VF)
Wanted View from the side (VS) of the solid
(A) (B)
(C)
(D)
Degree of difficulty high
Question type 2
The following items also test your ability to visualise three-dimensional figures Each item consists of two illustrations showing a transparent cube with one or two cables in its interior The first illustration (left) always shows the view from the front In the picture on the right the same cube is illustrated again Your task is to determine whether the picture on the right shows that cube from the right (r) left (l) from below (w) above (a) or behind (d)
(A) r (B) l (C) w (D) a (E) d
Here you see the cube from the front
Here you see the cube from
In the picture on the right you see the cube from above On your answer sheet you would mark the (D)
These items can be solved in one of the following two ways
Imagine that the cube had been placed on a glass table and that you could walk all the way around it Standing to the right or left of the table you look at the cube from the right or from the left If you go behind the table you look at the cube from behind If you come back to the front of the table and bend over it to look at the cube from above and if you imagine yourself lying down underneath the table feet first you see the view from below
Or you imagine that you could pick up the cube and turn it around in your hands If you looked at the cube from the front ie from the position shown in the left-hand illustration and then tipped it towards you by 90 degrees not changing your own position at all then you would see the view from above If you looked at the cube from the front and then turned it 90 degrees to the right you would see the view from the left If you turned it from the starting position 90 degrees to the left you would see it from the right And if you turned it 180 degrees to the right or left from the starting position you would see it from behind Finally if you tipped it backward you would see it from below
Example 1
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty low Example 2
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty medium
Example 3
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front
Here you see the cube from
Degree of difficulty high
Solutions Subtest rdquoVisualising Solidsldquo Question type 1 Example 1 Visualise this solid as a tree stump which has been cut off diagonally When you look at it from above (view from above) you see that a fairly large piece has been cut out of its left half Behind the cut-out section however a relatively large section of the stump has remained standing You can therefore rule out Option (A) immediately because this option shows nothing remaining behind the cut-out section except for the outer bark Option (B) can likewise not be the correct solution since here only a section of bark has been removed from the front The piece cut out of Option (D) has only one straight side According to the view from above however the cut-out section has to have three straight sides This is the case only with the section cut out of Option (C) Therefore (C) is the correct answer Example 2 One means of solving this item is to begin by looking at a detail which does not occur in all four answer options Look for example at the figure which looks like an upside-down L on the left edge of Option (B) and (D) Does this figure result from the view from above and the view from the front Yes it does because the upside-down L is the unobstructed view of the high surface at the figurersquos centre Accordingly you can already rule out Options (A) and (C) Options (B) und (D) differ in that in Figure (B) a step has been indicated over the rectangular figure to the right whereas (D) shows a straight edge all the way to the top In the view from above however you can see the protruding section which forms the step Therefore (B) is the correct answer here
Example 3 In the process of solving this item it is initially unclear whether the side view we are looking for is a view from the left or one from the right On the basis of the view from above it can be deduced that in the view from the left side (VSL) one of the two protruding beams at the upper end of the figure points towards the viewer the other towards the right In the view from the right side (VSR) only one of the two protruding beams can be seen it points towards the left The answers (B) and (D) are therefore options for the VSL and the answers (A) and (C) are options for the VSR Answer (B) is out of the question because there is no line indicating an edge at the transition from the figurersquos base to the vertical element or ldquopillarrdquo on top of it ndash On the basis of the view from above it can be deduced that the base of the figure is square Therefore any view from the side would have to indicate the upper edge of the base just as the view from the front does In the case of answer (D) there is a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam In the view from above however this line cannot be accounted for In the case of answer (A) on the other hand there is likewise a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam Here that line is correct because this is one of the two options for a VSR The left vertical edge of the pillar must therefore be visible in front of the protruding beam The other elements of answer (A) also correspond with the view of the three-dimensional figure from above and the view of it from the side and (A) is accordingly the correct solution to this task Answer (C) cannot be the correct solution because the transition between the upper end of the pillar and the upper protruding beam is shown as a single plane According to the view from above however the vertical edge of the uppermost end of the pillar would have to be visible here Question type 2 Example 1 In the case of this simple example you can immediately rule out the perspective from below and above From below as well as from above you would be looking through a kind of tube Therefore the perspective illustrated on the right can only be the view from the right the left or behind Now look at the bottom end of the cable In the left-hand picture it faces you In the right-hand picture it faces away from you ie points in exactly the opposite direction Therefore it is clear that the right-hand picture shows the view from behind
Example 2 Here the only view you can rule out immediately is the one from behind (Option E) If the view from the front shows one end of the cable leading toward the back of the cube at the top right the view from behind would show this cable end coming at you at the top left This is not the case in the right-hand picture If you tip the cube forward in your imagination you immediately see that the correct answer cannot be the view from above and turning the cube 180 degrees or 90 degrees to the right also does not lead to the desired perspective But if you imagine yourself standing on the right side of the cube you see that the end of the cable which is concealed in the left-hand illustration comes toward you on the right side of the cube in the right-hand illustration Therefore right (A) is the correct answer
Example 3 Here the figure on the right cannot be showing the view of the cable from the left (B) In the view from the left the section of the cable running horizontally in the view from the front would have to be visible in the middle of the right-hand edge Answer (E) is incorrect for the same reason In the view from behind the horizontal section of cable would have to be
visible in the background likewise running horizontally about halfway between bottom and top
In the view from above (D) this same section of cable would have to be seen leading from one side to the other along the bottom surface of the cube The figure on the right cannot be showing the view from the right (A) because the part of the cable touching the upper left-hand wall in that figure would have to be touching the upper right-hand wall in the view from the front which is not the case The only remaining option is the view from below (C) which only reveals itself to be correct however upon closer examination We might easily find ourselves looking for the end of the cable clearly seen at the bottom left in the view from the front ndash In the view from below it runs right into a curve in the cable and thus appears not to be an end at all On the other hand the end of the cable visible on the right-hand edge in the view from below is not visible in the view from the front because it is hidden behind a curving section of cable
In the subtest Analysing Technical Interrelationships you have to analyse and interpret diagrams charts or tables depicting technical laws or formulae
The test measures the ability to abstract from scientific and technical facts and to put abstract facts in concrete terms Knowledge of mathematics physics or technology is not needed background information will be provided if necessary
Instructions Working time 60 minutes
These items contain questions from various technical areas Your task is to visualize simple technical procedures and recognize technical interrelationships Unless otherwise indicated the axes (scales) of all diagrams are linearly subdivided In some of the items you must identify the ldquoqualitativelyrdquo correct diagram In other words your task is to decide which graph best represents the relationship between the variables shown Even the correct diagram will not necessarily drawn to scale
Example 1 Arrangements I and II each include a beam which is pivot-mounted (like a swing or see-saw) A hook has been mounted on the right end of the beam The ends of the beam are connected by a rope which is threaded through rolls
beam beam
bearing hook
rope rope
hookbearing
A weight is hung on the hook Which of the following statements isare then correct (The masses of the beam rope and hook can be neglected)
I In the case of Arrangement I the right end of the beam moves downward II In the case of Arrangement II the right end of the beam moves downward
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty low
Example 2 This diagram shows the wattage required by a lift motor in a period of 12 minutes (min) When the lift travels upward four times as much wattage (per minute) is required as when the lift travels downward When the lift stops at a floor twice as much wattage (per minute) is required as when the lift travels downward The running time between two consecutive floors is 30 seconds At the point in time t = 0 the lift is on the third floor
Which of the following statements isare correct
I Within the 12 minutes shown the lift travels up to the 6th floor II At the point in time t = 10 min the lift is on the 3rd floor
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty medium Example 3 The diagram shows two thermometers on which no temperature scales have yet been indicated They are both filled with the same liquid and the amount of liquid is also the same Their tubes are of the same length However the tube of the left-hand thermometer has a narrower diameter than that of the right-hand thermometer
We will assume that the markings for the two temperature scales are added correctly they begin at the same height on each tube and end at the same height Both thermometers are used only at temperatures for which they are suitable
Which of the two statements is (or are) therefore correct
I Rises in temperature can be measured less accurately with the left-hand thermometer than with the right-hand one II The right-hand thermometer covers a greater temperature range than the left-hand one
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty high
Solutions Subtest ldquoAnalysing Technical Interrelationshipsrdquo Example 1 If there is no rope the right end of the beam moves downward in both cases when a weight is suspended from the hook So the question is whether this movement is prevented by the rope When a weight is suspended from the hook in Arrangement I the rope slackens at the right end and at the left end The right end of the beam moves downward and Statement I is accordingly correct When a weight is suspended from the hook in Arrangement II traction (pulling power) is applied to the right end of the rope By way of the rope this traction is transferred to the left end of the beam Since both ends of the beam are pulled downward with the same force the beam does not move Statement II is false The solution to this item is therefore A
Example 2 Three different values for the wattage P are shown in the diagram 1 2 and 4 According to the text P is the lowest when the lift travels downward In this case therefore P = 1 When the lift stops at a floor P = 2 When the lift travels upward P = 4
On the basis of this information it is possible to interpret the movement of the lift At the point in time t = 0 the lift is on the 3rd floor and stops there for 1 minute Then it travels downward for 1 minute Since it takes 30 seconds to travel one floor it is then on the 1st floor Following a stop of 1 minute it travels upward 2 minutes (corresponding to 4 floors) At the point in time t = 5 it is therefore on the 5th floor There it stops for 15 minutes then travelling upward to the 6th floor ndash Statement I is therefore correct One minute later it travels 15 minutes (corresponding to 3 floors) downward and at the point in time t = 95 is therefore on the 3rd floor ndash Statement II is accordingly also correct
The solution to this item is therefore C
Example 3 If the temperature is increased by x degrees the liquid inside each thermometer expands by the same volume However this increase in liquid volume makes the liquid rise to a higher level in the tube of the thermometer on the left Since the cross-section of the tube in the left-hand thermometer is smaller than that of the right-hand one a defined temperature change generally leads here to a greater change in the liquid level than with the right-hand
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
The solution to this item is therefore B
Accordingly if
Subtest ldquoVisualising Solidsldquo
In the subtest Visualising Solids you have to infer perspectives of a solid from one given view of the solid The test measures your spatial sense Instructions Two parts Working time 30 minutes Question type 1
To solve the following items you are to visualize the bodies three-dimensionally In each exercise the body is shown from two perspectives You are to identify the view of the same body from a third perspective Please mark the correct solution (A B C or D) on your answer sheet
The viewsperspectives are referred to as follows
Parallel projection of a cube
Further pointers ndash In the illustrations all visible edges are depicted as continuous (uninterrupted) lines
ndash If the illustration of a view from the side is not accompanied by an arrow rarr indicating which of the two side views is intended part of the task is to find that out
ndash If for example a side view is illustrated to the right of the view from the front or the view from above it does not necessarily mean that it is a view from the right side
View from above (VA)
View from the front (VF)
View from the side (VS)
VA
VSVS(not visible
here)
Example 1 Given The view of a solid from above and one side view of the same solid
View from above (VA) View from one side (VS)
uarr
(VF)
Wanted View from the front (VF) of the solid
(A) (B)
(C) (D)
Degree of difficulty low
Example 2 Given The view of a solid from above and the view from the front of the same solid
View from above (VA) View from the front (VF)
uarr (VS)
Wanted View (VS) from the side of the solid indicated by the arrow
(A) (B)
(C) (D)
Degree of difficulty medium
Example 3 Given The View of a solid from above and the view from the front of the some solid
View from above (VA) View from the front (VF)
uarr (VF)
Wanted View from the side (VS) of the solid
(A) (B)
(C)
(D)
Degree of difficulty high
Question type 2
The following items also test your ability to visualise three-dimensional figures Each item consists of two illustrations showing a transparent cube with one or two cables in its interior The first illustration (left) always shows the view from the front In the picture on the right the same cube is illustrated again Your task is to determine whether the picture on the right shows that cube from the right (r) left (l) from below (w) above (a) or behind (d)
(A) r (B) l (C) w (D) a (E) d
Here you see the cube from the front
Here you see the cube from
In the picture on the right you see the cube from above On your answer sheet you would mark the (D)
These items can be solved in one of the following two ways
Imagine that the cube had been placed on a glass table and that you could walk all the way around it Standing to the right or left of the table you look at the cube from the right or from the left If you go behind the table you look at the cube from behind If you come back to the front of the table and bend over it to look at the cube from above and if you imagine yourself lying down underneath the table feet first you see the view from below
Or you imagine that you could pick up the cube and turn it around in your hands If you looked at the cube from the front ie from the position shown in the left-hand illustration and then tipped it towards you by 90 degrees not changing your own position at all then you would see the view from above If you looked at the cube from the front and then turned it 90 degrees to the right you would see the view from the left If you turned it from the starting position 90 degrees to the left you would see it from the right And if you turned it 180 degrees to the right or left from the starting position you would see it from behind Finally if you tipped it backward you would see it from below
Example 1
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty low Example 2
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty medium
Example 3
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front
Here you see the cube from
Degree of difficulty high
Solutions Subtest rdquoVisualising Solidsldquo Question type 1 Example 1 Visualise this solid as a tree stump which has been cut off diagonally When you look at it from above (view from above) you see that a fairly large piece has been cut out of its left half Behind the cut-out section however a relatively large section of the stump has remained standing You can therefore rule out Option (A) immediately because this option shows nothing remaining behind the cut-out section except for the outer bark Option (B) can likewise not be the correct solution since here only a section of bark has been removed from the front The piece cut out of Option (D) has only one straight side According to the view from above however the cut-out section has to have three straight sides This is the case only with the section cut out of Option (C) Therefore (C) is the correct answer Example 2 One means of solving this item is to begin by looking at a detail which does not occur in all four answer options Look for example at the figure which looks like an upside-down L on the left edge of Option (B) and (D) Does this figure result from the view from above and the view from the front Yes it does because the upside-down L is the unobstructed view of the high surface at the figurersquos centre Accordingly you can already rule out Options (A) and (C) Options (B) und (D) differ in that in Figure (B) a step has been indicated over the rectangular figure to the right whereas (D) shows a straight edge all the way to the top In the view from above however you can see the protruding section which forms the step Therefore (B) is the correct answer here
Example 3 In the process of solving this item it is initially unclear whether the side view we are looking for is a view from the left or one from the right On the basis of the view from above it can be deduced that in the view from the left side (VSL) one of the two protruding beams at the upper end of the figure points towards the viewer the other towards the right In the view from the right side (VSR) only one of the two protruding beams can be seen it points towards the left The answers (B) and (D) are therefore options for the VSL and the answers (A) and (C) are options for the VSR Answer (B) is out of the question because there is no line indicating an edge at the transition from the figurersquos base to the vertical element or ldquopillarrdquo on top of it ndash On the basis of the view from above it can be deduced that the base of the figure is square Therefore any view from the side would have to indicate the upper edge of the base just as the view from the front does In the case of answer (D) there is a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam In the view from above however this line cannot be accounted for In the case of answer (A) on the other hand there is likewise a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam Here that line is correct because this is one of the two options for a VSR The left vertical edge of the pillar must therefore be visible in front of the protruding beam The other elements of answer (A) also correspond with the view of the three-dimensional figure from above and the view of it from the side and (A) is accordingly the correct solution to this task Answer (C) cannot be the correct solution because the transition between the upper end of the pillar and the upper protruding beam is shown as a single plane According to the view from above however the vertical edge of the uppermost end of the pillar would have to be visible here Question type 2 Example 1 In the case of this simple example you can immediately rule out the perspective from below and above From below as well as from above you would be looking through a kind of tube Therefore the perspective illustrated on the right can only be the view from the right the left or behind Now look at the bottom end of the cable In the left-hand picture it faces you In the right-hand picture it faces away from you ie points in exactly the opposite direction Therefore it is clear that the right-hand picture shows the view from behind
Example 2 Here the only view you can rule out immediately is the one from behind (Option E) If the view from the front shows one end of the cable leading toward the back of the cube at the top right the view from behind would show this cable end coming at you at the top left This is not the case in the right-hand picture If you tip the cube forward in your imagination you immediately see that the correct answer cannot be the view from above and turning the cube 180 degrees or 90 degrees to the right also does not lead to the desired perspective But if you imagine yourself standing on the right side of the cube you see that the end of the cable which is concealed in the left-hand illustration comes toward you on the right side of the cube in the right-hand illustration Therefore right (A) is the correct answer
Example 3 Here the figure on the right cannot be showing the view of the cable from the left (B) In the view from the left the section of the cable running horizontally in the view from the front would have to be visible in the middle of the right-hand edge Answer (E) is incorrect for the same reason In the view from behind the horizontal section of cable would have to be
visible in the background likewise running horizontally about halfway between bottom and top
In the view from above (D) this same section of cable would have to be seen leading from one side to the other along the bottom surface of the cube The figure on the right cannot be showing the view from the right (A) because the part of the cable touching the upper left-hand wall in that figure would have to be touching the upper right-hand wall in the view from the front which is not the case The only remaining option is the view from below (C) which only reveals itself to be correct however upon closer examination We might easily find ourselves looking for the end of the cable clearly seen at the bottom left in the view from the front ndash In the view from below it runs right into a curve in the cable and thus appears not to be an end at all On the other hand the end of the cable visible on the right-hand edge in the view from below is not visible in the view from the front because it is hidden behind a curving section of cable
In the subtest Analysing Technical Interrelationships you have to analyse and interpret diagrams charts or tables depicting technical laws or formulae
The test measures the ability to abstract from scientific and technical facts and to put abstract facts in concrete terms Knowledge of mathematics physics or technology is not needed background information will be provided if necessary
Instructions Working time 60 minutes
These items contain questions from various technical areas Your task is to visualize simple technical procedures and recognize technical interrelationships Unless otherwise indicated the axes (scales) of all diagrams are linearly subdivided In some of the items you must identify the ldquoqualitativelyrdquo correct diagram In other words your task is to decide which graph best represents the relationship between the variables shown Even the correct diagram will not necessarily drawn to scale
Example 1 Arrangements I and II each include a beam which is pivot-mounted (like a swing or see-saw) A hook has been mounted on the right end of the beam The ends of the beam are connected by a rope which is threaded through rolls
beam beam
bearing hook
rope rope
hookbearing
A weight is hung on the hook Which of the following statements isare then correct (The masses of the beam rope and hook can be neglected)
I In the case of Arrangement I the right end of the beam moves downward II In the case of Arrangement II the right end of the beam moves downward
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty low
Example 2 This diagram shows the wattage required by a lift motor in a period of 12 minutes (min) When the lift travels upward four times as much wattage (per minute) is required as when the lift travels downward When the lift stops at a floor twice as much wattage (per minute) is required as when the lift travels downward The running time between two consecutive floors is 30 seconds At the point in time t = 0 the lift is on the third floor
Which of the following statements isare correct
I Within the 12 minutes shown the lift travels up to the 6th floor II At the point in time t = 10 min the lift is on the 3rd floor
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty medium Example 3 The diagram shows two thermometers on which no temperature scales have yet been indicated They are both filled with the same liquid and the amount of liquid is also the same Their tubes are of the same length However the tube of the left-hand thermometer has a narrower diameter than that of the right-hand thermometer
We will assume that the markings for the two temperature scales are added correctly they begin at the same height on each tube and end at the same height Both thermometers are used only at temperatures for which they are suitable
Which of the two statements is (or are) therefore correct
I Rises in temperature can be measured less accurately with the left-hand thermometer than with the right-hand one II The right-hand thermometer covers a greater temperature range than the left-hand one
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty high
Solutions Subtest ldquoAnalysing Technical Interrelationshipsrdquo Example 1 If there is no rope the right end of the beam moves downward in both cases when a weight is suspended from the hook So the question is whether this movement is prevented by the rope When a weight is suspended from the hook in Arrangement I the rope slackens at the right end and at the left end The right end of the beam moves downward and Statement I is accordingly correct When a weight is suspended from the hook in Arrangement II traction (pulling power) is applied to the right end of the rope By way of the rope this traction is transferred to the left end of the beam Since both ends of the beam are pulled downward with the same force the beam does not move Statement II is false The solution to this item is therefore A
Example 2 Three different values for the wattage P are shown in the diagram 1 2 and 4 According to the text P is the lowest when the lift travels downward In this case therefore P = 1 When the lift stops at a floor P = 2 When the lift travels upward P = 4
On the basis of this information it is possible to interpret the movement of the lift At the point in time t = 0 the lift is on the 3rd floor and stops there for 1 minute Then it travels downward for 1 minute Since it takes 30 seconds to travel one floor it is then on the 1st floor Following a stop of 1 minute it travels upward 2 minutes (corresponding to 4 floors) At the point in time t = 5 it is therefore on the 5th floor There it stops for 15 minutes then travelling upward to the 6th floor ndash Statement I is therefore correct One minute later it travels 15 minutes (corresponding to 3 floors) downward and at the point in time t = 95 is therefore on the 3rd floor ndash Statement II is accordingly also correct
The solution to this item is therefore C
Example 3 If the temperature is increased by x degrees the liquid inside each thermometer expands by the same volume However this increase in liquid volume makes the liquid rise to a higher level in the tube of the thermometer on the left Since the cross-section of the tube in the left-hand thermometer is smaller than that of the right-hand one a defined temperature change generally leads here to a greater change in the liquid level than with the right-hand
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
The solution to this item is therefore B
Subtest ldquoVisualising Solidsldquo
In the subtest Visualising Solids you have to infer perspectives of a solid from one given view of the solid The test measures your spatial sense Instructions Two parts Working time 30 minutes Question type 1
To solve the following items you are to visualize the bodies three-dimensionally In each exercise the body is shown from two perspectives You are to identify the view of the same body from a third perspective Please mark the correct solution (A B C or D) on your answer sheet
The viewsperspectives are referred to as follows
Parallel projection of a cube
Further pointers ndash In the illustrations all visible edges are depicted as continuous (uninterrupted) lines
ndash If the illustration of a view from the side is not accompanied by an arrow rarr indicating which of the two side views is intended part of the task is to find that out
ndash If for example a side view is illustrated to the right of the view from the front or the view from above it does not necessarily mean that it is a view from the right side
View from above (VA)
View from the front (VF)
View from the side (VS)
VA
VSVS(not visible
here)
Example 1 Given The view of a solid from above and one side view of the same solid
View from above (VA) View from one side (VS)
uarr
(VF)
Wanted View from the front (VF) of the solid
(A) (B)
(C) (D)
Degree of difficulty low
Example 2 Given The view of a solid from above and the view from the front of the same solid
View from above (VA) View from the front (VF)
uarr (VS)
Wanted View (VS) from the side of the solid indicated by the arrow
(A) (B)
(C) (D)
Degree of difficulty medium
Example 3 Given The View of a solid from above and the view from the front of the some solid
View from above (VA) View from the front (VF)
uarr (VF)
Wanted View from the side (VS) of the solid
(A) (B)
(C)
(D)
Degree of difficulty high
Question type 2
The following items also test your ability to visualise three-dimensional figures Each item consists of two illustrations showing a transparent cube with one or two cables in its interior The first illustration (left) always shows the view from the front In the picture on the right the same cube is illustrated again Your task is to determine whether the picture on the right shows that cube from the right (r) left (l) from below (w) above (a) or behind (d)
(A) r (B) l (C) w (D) a (E) d
Here you see the cube from the front
Here you see the cube from
In the picture on the right you see the cube from above On your answer sheet you would mark the (D)
These items can be solved in one of the following two ways
Imagine that the cube had been placed on a glass table and that you could walk all the way around it Standing to the right or left of the table you look at the cube from the right or from the left If you go behind the table you look at the cube from behind If you come back to the front of the table and bend over it to look at the cube from above and if you imagine yourself lying down underneath the table feet first you see the view from below
Or you imagine that you could pick up the cube and turn it around in your hands If you looked at the cube from the front ie from the position shown in the left-hand illustration and then tipped it towards you by 90 degrees not changing your own position at all then you would see the view from above If you looked at the cube from the front and then turned it 90 degrees to the right you would see the view from the left If you turned it from the starting position 90 degrees to the left you would see it from the right And if you turned it 180 degrees to the right or left from the starting position you would see it from behind Finally if you tipped it backward you would see it from below
Example 1
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty low Example 2
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty medium
Example 3
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front
Here you see the cube from
Degree of difficulty high
Solutions Subtest rdquoVisualising Solidsldquo Question type 1 Example 1 Visualise this solid as a tree stump which has been cut off diagonally When you look at it from above (view from above) you see that a fairly large piece has been cut out of its left half Behind the cut-out section however a relatively large section of the stump has remained standing You can therefore rule out Option (A) immediately because this option shows nothing remaining behind the cut-out section except for the outer bark Option (B) can likewise not be the correct solution since here only a section of bark has been removed from the front The piece cut out of Option (D) has only one straight side According to the view from above however the cut-out section has to have three straight sides This is the case only with the section cut out of Option (C) Therefore (C) is the correct answer Example 2 One means of solving this item is to begin by looking at a detail which does not occur in all four answer options Look for example at the figure which looks like an upside-down L on the left edge of Option (B) and (D) Does this figure result from the view from above and the view from the front Yes it does because the upside-down L is the unobstructed view of the high surface at the figurersquos centre Accordingly you can already rule out Options (A) and (C) Options (B) und (D) differ in that in Figure (B) a step has been indicated over the rectangular figure to the right whereas (D) shows a straight edge all the way to the top In the view from above however you can see the protruding section which forms the step Therefore (B) is the correct answer here
Example 3 In the process of solving this item it is initially unclear whether the side view we are looking for is a view from the left or one from the right On the basis of the view from above it can be deduced that in the view from the left side (VSL) one of the two protruding beams at the upper end of the figure points towards the viewer the other towards the right In the view from the right side (VSR) only one of the two protruding beams can be seen it points towards the left The answers (B) and (D) are therefore options for the VSL and the answers (A) and (C) are options for the VSR Answer (B) is out of the question because there is no line indicating an edge at the transition from the figurersquos base to the vertical element or ldquopillarrdquo on top of it ndash On the basis of the view from above it can be deduced that the base of the figure is square Therefore any view from the side would have to indicate the upper edge of the base just as the view from the front does In the case of answer (D) there is a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam In the view from above however this line cannot be accounted for In the case of answer (A) on the other hand there is likewise a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam Here that line is correct because this is one of the two options for a VSR The left vertical edge of the pillar must therefore be visible in front of the protruding beam The other elements of answer (A) also correspond with the view of the three-dimensional figure from above and the view of it from the side and (A) is accordingly the correct solution to this task Answer (C) cannot be the correct solution because the transition between the upper end of the pillar and the upper protruding beam is shown as a single plane According to the view from above however the vertical edge of the uppermost end of the pillar would have to be visible here Question type 2 Example 1 In the case of this simple example you can immediately rule out the perspective from below and above From below as well as from above you would be looking through a kind of tube Therefore the perspective illustrated on the right can only be the view from the right the left or behind Now look at the bottom end of the cable In the left-hand picture it faces you In the right-hand picture it faces away from you ie points in exactly the opposite direction Therefore it is clear that the right-hand picture shows the view from behind
Example 2 Here the only view you can rule out immediately is the one from behind (Option E) If the view from the front shows one end of the cable leading toward the back of the cube at the top right the view from behind would show this cable end coming at you at the top left This is not the case in the right-hand picture If you tip the cube forward in your imagination you immediately see that the correct answer cannot be the view from above and turning the cube 180 degrees or 90 degrees to the right also does not lead to the desired perspective But if you imagine yourself standing on the right side of the cube you see that the end of the cable which is concealed in the left-hand illustration comes toward you on the right side of the cube in the right-hand illustration Therefore right (A) is the correct answer
Example 3 Here the figure on the right cannot be showing the view of the cable from the left (B) In the view from the left the section of the cable running horizontally in the view from the front would have to be visible in the middle of the right-hand edge Answer (E) is incorrect for the same reason In the view from behind the horizontal section of cable would have to be
visible in the background likewise running horizontally about halfway between bottom and top
In the view from above (D) this same section of cable would have to be seen leading from one side to the other along the bottom surface of the cube The figure on the right cannot be showing the view from the right (A) because the part of the cable touching the upper left-hand wall in that figure would have to be touching the upper right-hand wall in the view from the front which is not the case The only remaining option is the view from below (C) which only reveals itself to be correct however upon closer examination We might easily find ourselves looking for the end of the cable clearly seen at the bottom left in the view from the front ndash In the view from below it runs right into a curve in the cable and thus appears not to be an end at all On the other hand the end of the cable visible on the right-hand edge in the view from below is not visible in the view from the front because it is hidden behind a curving section of cable
In the subtest Analysing Technical Interrelationships you have to analyse and interpret diagrams charts or tables depicting technical laws or formulae
The test measures the ability to abstract from scientific and technical facts and to put abstract facts in concrete terms Knowledge of mathematics physics or technology is not needed background information will be provided if necessary
Instructions Working time 60 minutes
These items contain questions from various technical areas Your task is to visualize simple technical procedures and recognize technical interrelationships Unless otherwise indicated the axes (scales) of all diagrams are linearly subdivided In some of the items you must identify the ldquoqualitativelyrdquo correct diagram In other words your task is to decide which graph best represents the relationship between the variables shown Even the correct diagram will not necessarily drawn to scale
Example 1 Arrangements I and II each include a beam which is pivot-mounted (like a swing or see-saw) A hook has been mounted on the right end of the beam The ends of the beam are connected by a rope which is threaded through rolls
beam beam
bearing hook
rope rope
hookbearing
A weight is hung on the hook Which of the following statements isare then correct (The masses of the beam rope and hook can be neglected)
I In the case of Arrangement I the right end of the beam moves downward II In the case of Arrangement II the right end of the beam moves downward
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty low
Example 2 This diagram shows the wattage required by a lift motor in a period of 12 minutes (min) When the lift travels upward four times as much wattage (per minute) is required as when the lift travels downward When the lift stops at a floor twice as much wattage (per minute) is required as when the lift travels downward The running time between two consecutive floors is 30 seconds At the point in time t = 0 the lift is on the third floor
Which of the following statements isare correct
I Within the 12 minutes shown the lift travels up to the 6th floor II At the point in time t = 10 min the lift is on the 3rd floor
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty medium Example 3 The diagram shows two thermometers on which no temperature scales have yet been indicated They are both filled with the same liquid and the amount of liquid is also the same Their tubes are of the same length However the tube of the left-hand thermometer has a narrower diameter than that of the right-hand thermometer
We will assume that the markings for the two temperature scales are added correctly they begin at the same height on each tube and end at the same height Both thermometers are used only at temperatures for which they are suitable
Which of the two statements is (or are) therefore correct
I Rises in temperature can be measured less accurately with the left-hand thermometer than with the right-hand one II The right-hand thermometer covers a greater temperature range than the left-hand one
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty high
Solutions Subtest ldquoAnalysing Technical Interrelationshipsrdquo Example 1 If there is no rope the right end of the beam moves downward in both cases when a weight is suspended from the hook So the question is whether this movement is prevented by the rope When a weight is suspended from the hook in Arrangement I the rope slackens at the right end and at the left end The right end of the beam moves downward and Statement I is accordingly correct When a weight is suspended from the hook in Arrangement II traction (pulling power) is applied to the right end of the rope By way of the rope this traction is transferred to the left end of the beam Since both ends of the beam are pulled downward with the same force the beam does not move Statement II is false The solution to this item is therefore A
Example 2 Three different values for the wattage P are shown in the diagram 1 2 and 4 According to the text P is the lowest when the lift travels downward In this case therefore P = 1 When the lift stops at a floor P = 2 When the lift travels upward P = 4
On the basis of this information it is possible to interpret the movement of the lift At the point in time t = 0 the lift is on the 3rd floor and stops there for 1 minute Then it travels downward for 1 minute Since it takes 30 seconds to travel one floor it is then on the 1st floor Following a stop of 1 minute it travels upward 2 minutes (corresponding to 4 floors) At the point in time t = 5 it is therefore on the 5th floor There it stops for 15 minutes then travelling upward to the 6th floor ndash Statement I is therefore correct One minute later it travels 15 minutes (corresponding to 3 floors) downward and at the point in time t = 95 is therefore on the 3rd floor ndash Statement II is accordingly also correct
The solution to this item is therefore C
Example 3 If the temperature is increased by x degrees the liquid inside each thermometer expands by the same volume However this increase in liquid volume makes the liquid rise to a higher level in the tube of the thermometer on the left Since the cross-section of the tube in the left-hand thermometer is smaller than that of the right-hand one a defined temperature change generally leads here to a greater change in the liquid level than with the right-hand
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
The solution to this item is therefore B
Example 1 Given The view of a solid from above and one side view of the same solid
View from above (VA) View from one side (VS)
uarr
(VF)
Wanted View from the front (VF) of the solid
(A) (B)
(C) (D)
Degree of difficulty low
Example 2 Given The view of a solid from above and the view from the front of the same solid
View from above (VA) View from the front (VF)
uarr (VS)
Wanted View (VS) from the side of the solid indicated by the arrow
(A) (B)
(C) (D)
Degree of difficulty medium
Example 3 Given The View of a solid from above and the view from the front of the some solid
View from above (VA) View from the front (VF)
uarr (VF)
Wanted View from the side (VS) of the solid
(A) (B)
(C)
(D)
Degree of difficulty high
Question type 2
The following items also test your ability to visualise three-dimensional figures Each item consists of two illustrations showing a transparent cube with one or two cables in its interior The first illustration (left) always shows the view from the front In the picture on the right the same cube is illustrated again Your task is to determine whether the picture on the right shows that cube from the right (r) left (l) from below (w) above (a) or behind (d)
(A) r (B) l (C) w (D) a (E) d
Here you see the cube from the front
Here you see the cube from
In the picture on the right you see the cube from above On your answer sheet you would mark the (D)
These items can be solved in one of the following two ways
Imagine that the cube had been placed on a glass table and that you could walk all the way around it Standing to the right or left of the table you look at the cube from the right or from the left If you go behind the table you look at the cube from behind If you come back to the front of the table and bend over it to look at the cube from above and if you imagine yourself lying down underneath the table feet first you see the view from below
Or you imagine that you could pick up the cube and turn it around in your hands If you looked at the cube from the front ie from the position shown in the left-hand illustration and then tipped it towards you by 90 degrees not changing your own position at all then you would see the view from above If you looked at the cube from the front and then turned it 90 degrees to the right you would see the view from the left If you turned it from the starting position 90 degrees to the left you would see it from the right And if you turned it 180 degrees to the right or left from the starting position you would see it from behind Finally if you tipped it backward you would see it from below
Example 1
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty low Example 2
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty medium
Example 3
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front
Here you see the cube from
Degree of difficulty high
Solutions Subtest rdquoVisualising Solidsldquo Question type 1 Example 1 Visualise this solid as a tree stump which has been cut off diagonally When you look at it from above (view from above) you see that a fairly large piece has been cut out of its left half Behind the cut-out section however a relatively large section of the stump has remained standing You can therefore rule out Option (A) immediately because this option shows nothing remaining behind the cut-out section except for the outer bark Option (B) can likewise not be the correct solution since here only a section of bark has been removed from the front The piece cut out of Option (D) has only one straight side According to the view from above however the cut-out section has to have three straight sides This is the case only with the section cut out of Option (C) Therefore (C) is the correct answer Example 2 One means of solving this item is to begin by looking at a detail which does not occur in all four answer options Look for example at the figure which looks like an upside-down L on the left edge of Option (B) and (D) Does this figure result from the view from above and the view from the front Yes it does because the upside-down L is the unobstructed view of the high surface at the figurersquos centre Accordingly you can already rule out Options (A) and (C) Options (B) und (D) differ in that in Figure (B) a step has been indicated over the rectangular figure to the right whereas (D) shows a straight edge all the way to the top In the view from above however you can see the protruding section which forms the step Therefore (B) is the correct answer here
Example 3 In the process of solving this item it is initially unclear whether the side view we are looking for is a view from the left or one from the right On the basis of the view from above it can be deduced that in the view from the left side (VSL) one of the two protruding beams at the upper end of the figure points towards the viewer the other towards the right In the view from the right side (VSR) only one of the two protruding beams can be seen it points towards the left The answers (B) and (D) are therefore options for the VSL and the answers (A) and (C) are options for the VSR Answer (B) is out of the question because there is no line indicating an edge at the transition from the figurersquos base to the vertical element or ldquopillarrdquo on top of it ndash On the basis of the view from above it can be deduced that the base of the figure is square Therefore any view from the side would have to indicate the upper edge of the base just as the view from the front does In the case of answer (D) there is a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam In the view from above however this line cannot be accounted for In the case of answer (A) on the other hand there is likewise a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam Here that line is correct because this is one of the two options for a VSR The left vertical edge of the pillar must therefore be visible in front of the protruding beam The other elements of answer (A) also correspond with the view of the three-dimensional figure from above and the view of it from the side and (A) is accordingly the correct solution to this task Answer (C) cannot be the correct solution because the transition between the upper end of the pillar and the upper protruding beam is shown as a single plane According to the view from above however the vertical edge of the uppermost end of the pillar would have to be visible here Question type 2 Example 1 In the case of this simple example you can immediately rule out the perspective from below and above From below as well as from above you would be looking through a kind of tube Therefore the perspective illustrated on the right can only be the view from the right the left or behind Now look at the bottom end of the cable In the left-hand picture it faces you In the right-hand picture it faces away from you ie points in exactly the opposite direction Therefore it is clear that the right-hand picture shows the view from behind
Example 2 Here the only view you can rule out immediately is the one from behind (Option E) If the view from the front shows one end of the cable leading toward the back of the cube at the top right the view from behind would show this cable end coming at you at the top left This is not the case in the right-hand picture If you tip the cube forward in your imagination you immediately see that the correct answer cannot be the view from above and turning the cube 180 degrees or 90 degrees to the right also does not lead to the desired perspective But if you imagine yourself standing on the right side of the cube you see that the end of the cable which is concealed in the left-hand illustration comes toward you on the right side of the cube in the right-hand illustration Therefore right (A) is the correct answer
Example 3 Here the figure on the right cannot be showing the view of the cable from the left (B) In the view from the left the section of the cable running horizontally in the view from the front would have to be visible in the middle of the right-hand edge Answer (E) is incorrect for the same reason In the view from behind the horizontal section of cable would have to be
visible in the background likewise running horizontally about halfway between bottom and top
In the view from above (D) this same section of cable would have to be seen leading from one side to the other along the bottom surface of the cube The figure on the right cannot be showing the view from the right (A) because the part of the cable touching the upper left-hand wall in that figure would have to be touching the upper right-hand wall in the view from the front which is not the case The only remaining option is the view from below (C) which only reveals itself to be correct however upon closer examination We might easily find ourselves looking for the end of the cable clearly seen at the bottom left in the view from the front ndash In the view from below it runs right into a curve in the cable and thus appears not to be an end at all On the other hand the end of the cable visible on the right-hand edge in the view from below is not visible in the view from the front because it is hidden behind a curving section of cable
In the subtest Analysing Technical Interrelationships you have to analyse and interpret diagrams charts or tables depicting technical laws or formulae
The test measures the ability to abstract from scientific and technical facts and to put abstract facts in concrete terms Knowledge of mathematics physics or technology is not needed background information will be provided if necessary
Instructions Working time 60 minutes
These items contain questions from various technical areas Your task is to visualize simple technical procedures and recognize technical interrelationships Unless otherwise indicated the axes (scales) of all diagrams are linearly subdivided In some of the items you must identify the ldquoqualitativelyrdquo correct diagram In other words your task is to decide which graph best represents the relationship between the variables shown Even the correct diagram will not necessarily drawn to scale
Example 1 Arrangements I and II each include a beam which is pivot-mounted (like a swing or see-saw) A hook has been mounted on the right end of the beam The ends of the beam are connected by a rope which is threaded through rolls
beam beam
bearing hook
rope rope
hookbearing
A weight is hung on the hook Which of the following statements isare then correct (The masses of the beam rope and hook can be neglected)
I In the case of Arrangement I the right end of the beam moves downward II In the case of Arrangement II the right end of the beam moves downward
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty low
Example 2 This diagram shows the wattage required by a lift motor in a period of 12 minutes (min) When the lift travels upward four times as much wattage (per minute) is required as when the lift travels downward When the lift stops at a floor twice as much wattage (per minute) is required as when the lift travels downward The running time between two consecutive floors is 30 seconds At the point in time t = 0 the lift is on the third floor
Which of the following statements isare correct
I Within the 12 minutes shown the lift travels up to the 6th floor II At the point in time t = 10 min the lift is on the 3rd floor
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty medium Example 3 The diagram shows two thermometers on which no temperature scales have yet been indicated They are both filled with the same liquid and the amount of liquid is also the same Their tubes are of the same length However the tube of the left-hand thermometer has a narrower diameter than that of the right-hand thermometer
We will assume that the markings for the two temperature scales are added correctly they begin at the same height on each tube and end at the same height Both thermometers are used only at temperatures for which they are suitable
Which of the two statements is (or are) therefore correct
I Rises in temperature can be measured less accurately with the left-hand thermometer than with the right-hand one II The right-hand thermometer covers a greater temperature range than the left-hand one
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty high
Solutions Subtest ldquoAnalysing Technical Interrelationshipsrdquo Example 1 If there is no rope the right end of the beam moves downward in both cases when a weight is suspended from the hook So the question is whether this movement is prevented by the rope When a weight is suspended from the hook in Arrangement I the rope slackens at the right end and at the left end The right end of the beam moves downward and Statement I is accordingly correct When a weight is suspended from the hook in Arrangement II traction (pulling power) is applied to the right end of the rope By way of the rope this traction is transferred to the left end of the beam Since both ends of the beam are pulled downward with the same force the beam does not move Statement II is false The solution to this item is therefore A
Example 2 Three different values for the wattage P are shown in the diagram 1 2 and 4 According to the text P is the lowest when the lift travels downward In this case therefore P = 1 When the lift stops at a floor P = 2 When the lift travels upward P = 4
On the basis of this information it is possible to interpret the movement of the lift At the point in time t = 0 the lift is on the 3rd floor and stops there for 1 minute Then it travels downward for 1 minute Since it takes 30 seconds to travel one floor it is then on the 1st floor Following a stop of 1 minute it travels upward 2 minutes (corresponding to 4 floors) At the point in time t = 5 it is therefore on the 5th floor There it stops for 15 minutes then travelling upward to the 6th floor ndash Statement I is therefore correct One minute later it travels 15 minutes (corresponding to 3 floors) downward and at the point in time t = 95 is therefore on the 3rd floor ndash Statement II is accordingly also correct
The solution to this item is therefore C
Example 3 If the temperature is increased by x degrees the liquid inside each thermometer expands by the same volume However this increase in liquid volume makes the liquid rise to a higher level in the tube of the thermometer on the left Since the cross-section of the tube in the left-hand thermometer is smaller than that of the right-hand one a defined temperature change generally leads here to a greater change in the liquid level than with the right-hand
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
The solution to this item is therefore B
Example 2 Given The view of a solid from above and the view from the front of the same solid
View from above (VA) View from the front (VF)
uarr (VS)
Wanted View (VS) from the side of the solid indicated by the arrow
(A) (B)
(C) (D)
Degree of difficulty medium
Example 3 Given The View of a solid from above and the view from the front of the some solid
View from above (VA) View from the front (VF)
uarr (VF)
Wanted View from the side (VS) of the solid
(A) (B)
(C)
(D)
Degree of difficulty high
Question type 2
The following items also test your ability to visualise three-dimensional figures Each item consists of two illustrations showing a transparent cube with one or two cables in its interior The first illustration (left) always shows the view from the front In the picture on the right the same cube is illustrated again Your task is to determine whether the picture on the right shows that cube from the right (r) left (l) from below (w) above (a) or behind (d)
(A) r (B) l (C) w (D) a (E) d
Here you see the cube from the front
Here you see the cube from
In the picture on the right you see the cube from above On your answer sheet you would mark the (D)
These items can be solved in one of the following two ways
Imagine that the cube had been placed on a glass table and that you could walk all the way around it Standing to the right or left of the table you look at the cube from the right or from the left If you go behind the table you look at the cube from behind If you come back to the front of the table and bend over it to look at the cube from above and if you imagine yourself lying down underneath the table feet first you see the view from below
Or you imagine that you could pick up the cube and turn it around in your hands If you looked at the cube from the front ie from the position shown in the left-hand illustration and then tipped it towards you by 90 degrees not changing your own position at all then you would see the view from above If you looked at the cube from the front and then turned it 90 degrees to the right you would see the view from the left If you turned it from the starting position 90 degrees to the left you would see it from the right And if you turned it 180 degrees to the right or left from the starting position you would see it from behind Finally if you tipped it backward you would see it from below
Example 1
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty low Example 2
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty medium
Example 3
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front
Here you see the cube from
Degree of difficulty high
Solutions Subtest rdquoVisualising Solidsldquo Question type 1 Example 1 Visualise this solid as a tree stump which has been cut off diagonally When you look at it from above (view from above) you see that a fairly large piece has been cut out of its left half Behind the cut-out section however a relatively large section of the stump has remained standing You can therefore rule out Option (A) immediately because this option shows nothing remaining behind the cut-out section except for the outer bark Option (B) can likewise not be the correct solution since here only a section of bark has been removed from the front The piece cut out of Option (D) has only one straight side According to the view from above however the cut-out section has to have three straight sides This is the case only with the section cut out of Option (C) Therefore (C) is the correct answer Example 2 One means of solving this item is to begin by looking at a detail which does not occur in all four answer options Look for example at the figure which looks like an upside-down L on the left edge of Option (B) and (D) Does this figure result from the view from above and the view from the front Yes it does because the upside-down L is the unobstructed view of the high surface at the figurersquos centre Accordingly you can already rule out Options (A) and (C) Options (B) und (D) differ in that in Figure (B) a step has been indicated over the rectangular figure to the right whereas (D) shows a straight edge all the way to the top In the view from above however you can see the protruding section which forms the step Therefore (B) is the correct answer here
Example 3 In the process of solving this item it is initially unclear whether the side view we are looking for is a view from the left or one from the right On the basis of the view from above it can be deduced that in the view from the left side (VSL) one of the two protruding beams at the upper end of the figure points towards the viewer the other towards the right In the view from the right side (VSR) only one of the two protruding beams can be seen it points towards the left The answers (B) and (D) are therefore options for the VSL and the answers (A) and (C) are options for the VSR Answer (B) is out of the question because there is no line indicating an edge at the transition from the figurersquos base to the vertical element or ldquopillarrdquo on top of it ndash On the basis of the view from above it can be deduced that the base of the figure is square Therefore any view from the side would have to indicate the upper edge of the base just as the view from the front does In the case of answer (D) there is a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam In the view from above however this line cannot be accounted for In the case of answer (A) on the other hand there is likewise a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam Here that line is correct because this is one of the two options for a VSR The left vertical edge of the pillar must therefore be visible in front of the protruding beam The other elements of answer (A) also correspond with the view of the three-dimensional figure from above and the view of it from the side and (A) is accordingly the correct solution to this task Answer (C) cannot be the correct solution because the transition between the upper end of the pillar and the upper protruding beam is shown as a single plane According to the view from above however the vertical edge of the uppermost end of the pillar would have to be visible here Question type 2 Example 1 In the case of this simple example you can immediately rule out the perspective from below and above From below as well as from above you would be looking through a kind of tube Therefore the perspective illustrated on the right can only be the view from the right the left or behind Now look at the bottom end of the cable In the left-hand picture it faces you In the right-hand picture it faces away from you ie points in exactly the opposite direction Therefore it is clear that the right-hand picture shows the view from behind
Example 2 Here the only view you can rule out immediately is the one from behind (Option E) If the view from the front shows one end of the cable leading toward the back of the cube at the top right the view from behind would show this cable end coming at you at the top left This is not the case in the right-hand picture If you tip the cube forward in your imagination you immediately see that the correct answer cannot be the view from above and turning the cube 180 degrees or 90 degrees to the right also does not lead to the desired perspective But if you imagine yourself standing on the right side of the cube you see that the end of the cable which is concealed in the left-hand illustration comes toward you on the right side of the cube in the right-hand illustration Therefore right (A) is the correct answer
Example 3 Here the figure on the right cannot be showing the view of the cable from the left (B) In the view from the left the section of the cable running horizontally in the view from the front would have to be visible in the middle of the right-hand edge Answer (E) is incorrect for the same reason In the view from behind the horizontal section of cable would have to be
visible in the background likewise running horizontally about halfway between bottom and top
In the view from above (D) this same section of cable would have to be seen leading from one side to the other along the bottom surface of the cube The figure on the right cannot be showing the view from the right (A) because the part of the cable touching the upper left-hand wall in that figure would have to be touching the upper right-hand wall in the view from the front which is not the case The only remaining option is the view from below (C) which only reveals itself to be correct however upon closer examination We might easily find ourselves looking for the end of the cable clearly seen at the bottom left in the view from the front ndash In the view from below it runs right into a curve in the cable and thus appears not to be an end at all On the other hand the end of the cable visible on the right-hand edge in the view from below is not visible in the view from the front because it is hidden behind a curving section of cable
In the subtest Analysing Technical Interrelationships you have to analyse and interpret diagrams charts or tables depicting technical laws or formulae
The test measures the ability to abstract from scientific and technical facts and to put abstract facts in concrete terms Knowledge of mathematics physics or technology is not needed background information will be provided if necessary
Instructions Working time 60 minutes
These items contain questions from various technical areas Your task is to visualize simple technical procedures and recognize technical interrelationships Unless otherwise indicated the axes (scales) of all diagrams are linearly subdivided In some of the items you must identify the ldquoqualitativelyrdquo correct diagram In other words your task is to decide which graph best represents the relationship between the variables shown Even the correct diagram will not necessarily drawn to scale
Example 1 Arrangements I and II each include a beam which is pivot-mounted (like a swing or see-saw) A hook has been mounted on the right end of the beam The ends of the beam are connected by a rope which is threaded through rolls
beam beam
bearing hook
rope rope
hookbearing
A weight is hung on the hook Which of the following statements isare then correct (The masses of the beam rope and hook can be neglected)
I In the case of Arrangement I the right end of the beam moves downward II In the case of Arrangement II the right end of the beam moves downward
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty low
Example 2 This diagram shows the wattage required by a lift motor in a period of 12 minutes (min) When the lift travels upward four times as much wattage (per minute) is required as when the lift travels downward When the lift stops at a floor twice as much wattage (per minute) is required as when the lift travels downward The running time between two consecutive floors is 30 seconds At the point in time t = 0 the lift is on the third floor
Which of the following statements isare correct
I Within the 12 minutes shown the lift travels up to the 6th floor II At the point in time t = 10 min the lift is on the 3rd floor
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty medium Example 3 The diagram shows two thermometers on which no temperature scales have yet been indicated They are both filled with the same liquid and the amount of liquid is also the same Their tubes are of the same length However the tube of the left-hand thermometer has a narrower diameter than that of the right-hand thermometer
We will assume that the markings for the two temperature scales are added correctly they begin at the same height on each tube and end at the same height Both thermometers are used only at temperatures for which they are suitable
Which of the two statements is (or are) therefore correct
I Rises in temperature can be measured less accurately with the left-hand thermometer than with the right-hand one II The right-hand thermometer covers a greater temperature range than the left-hand one
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty high
Solutions Subtest ldquoAnalysing Technical Interrelationshipsrdquo Example 1 If there is no rope the right end of the beam moves downward in both cases when a weight is suspended from the hook So the question is whether this movement is prevented by the rope When a weight is suspended from the hook in Arrangement I the rope slackens at the right end and at the left end The right end of the beam moves downward and Statement I is accordingly correct When a weight is suspended from the hook in Arrangement II traction (pulling power) is applied to the right end of the rope By way of the rope this traction is transferred to the left end of the beam Since both ends of the beam are pulled downward with the same force the beam does not move Statement II is false The solution to this item is therefore A
Example 2 Three different values for the wattage P are shown in the diagram 1 2 and 4 According to the text P is the lowest when the lift travels downward In this case therefore P = 1 When the lift stops at a floor P = 2 When the lift travels upward P = 4
On the basis of this information it is possible to interpret the movement of the lift At the point in time t = 0 the lift is on the 3rd floor and stops there for 1 minute Then it travels downward for 1 minute Since it takes 30 seconds to travel one floor it is then on the 1st floor Following a stop of 1 minute it travels upward 2 minutes (corresponding to 4 floors) At the point in time t = 5 it is therefore on the 5th floor There it stops for 15 minutes then travelling upward to the 6th floor ndash Statement I is therefore correct One minute later it travels 15 minutes (corresponding to 3 floors) downward and at the point in time t = 95 is therefore on the 3rd floor ndash Statement II is accordingly also correct
The solution to this item is therefore C
Example 3 If the temperature is increased by x degrees the liquid inside each thermometer expands by the same volume However this increase in liquid volume makes the liquid rise to a higher level in the tube of the thermometer on the left Since the cross-section of the tube in the left-hand thermometer is smaller than that of the right-hand one a defined temperature change generally leads here to a greater change in the liquid level than with the right-hand
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
The solution to this item is therefore B
Example 3 Given The View of a solid from above and the view from the front of the some solid
View from above (VA) View from the front (VF)
uarr (VF)
Wanted View from the side (VS) of the solid
(A) (B)
(C)
(D)
Degree of difficulty high
Question type 2
The following items also test your ability to visualise three-dimensional figures Each item consists of two illustrations showing a transparent cube with one or two cables in its interior The first illustration (left) always shows the view from the front In the picture on the right the same cube is illustrated again Your task is to determine whether the picture on the right shows that cube from the right (r) left (l) from below (w) above (a) or behind (d)
(A) r (B) l (C) w (D) a (E) d
Here you see the cube from the front
Here you see the cube from
In the picture on the right you see the cube from above On your answer sheet you would mark the (D)
These items can be solved in one of the following two ways
Imagine that the cube had been placed on a glass table and that you could walk all the way around it Standing to the right or left of the table you look at the cube from the right or from the left If you go behind the table you look at the cube from behind If you come back to the front of the table and bend over it to look at the cube from above and if you imagine yourself lying down underneath the table feet first you see the view from below
Or you imagine that you could pick up the cube and turn it around in your hands If you looked at the cube from the front ie from the position shown in the left-hand illustration and then tipped it towards you by 90 degrees not changing your own position at all then you would see the view from above If you looked at the cube from the front and then turned it 90 degrees to the right you would see the view from the left If you turned it from the starting position 90 degrees to the left you would see it from the right And if you turned it 180 degrees to the right or left from the starting position you would see it from behind Finally if you tipped it backward you would see it from below
Example 1
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty low Example 2
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty medium
Example 3
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front
Here you see the cube from
Degree of difficulty high
Solutions Subtest rdquoVisualising Solidsldquo Question type 1 Example 1 Visualise this solid as a tree stump which has been cut off diagonally When you look at it from above (view from above) you see that a fairly large piece has been cut out of its left half Behind the cut-out section however a relatively large section of the stump has remained standing You can therefore rule out Option (A) immediately because this option shows nothing remaining behind the cut-out section except for the outer bark Option (B) can likewise not be the correct solution since here only a section of bark has been removed from the front The piece cut out of Option (D) has only one straight side According to the view from above however the cut-out section has to have three straight sides This is the case only with the section cut out of Option (C) Therefore (C) is the correct answer Example 2 One means of solving this item is to begin by looking at a detail which does not occur in all four answer options Look for example at the figure which looks like an upside-down L on the left edge of Option (B) and (D) Does this figure result from the view from above and the view from the front Yes it does because the upside-down L is the unobstructed view of the high surface at the figurersquos centre Accordingly you can already rule out Options (A) and (C) Options (B) und (D) differ in that in Figure (B) a step has been indicated over the rectangular figure to the right whereas (D) shows a straight edge all the way to the top In the view from above however you can see the protruding section which forms the step Therefore (B) is the correct answer here
Example 3 In the process of solving this item it is initially unclear whether the side view we are looking for is a view from the left or one from the right On the basis of the view from above it can be deduced that in the view from the left side (VSL) one of the two protruding beams at the upper end of the figure points towards the viewer the other towards the right In the view from the right side (VSR) only one of the two protruding beams can be seen it points towards the left The answers (B) and (D) are therefore options for the VSL and the answers (A) and (C) are options for the VSR Answer (B) is out of the question because there is no line indicating an edge at the transition from the figurersquos base to the vertical element or ldquopillarrdquo on top of it ndash On the basis of the view from above it can be deduced that the base of the figure is square Therefore any view from the side would have to indicate the upper edge of the base just as the view from the front does In the case of answer (D) there is a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam In the view from above however this line cannot be accounted for In the case of answer (A) on the other hand there is likewise a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam Here that line is correct because this is one of the two options for a VSR The left vertical edge of the pillar must therefore be visible in front of the protruding beam The other elements of answer (A) also correspond with the view of the three-dimensional figure from above and the view of it from the side and (A) is accordingly the correct solution to this task Answer (C) cannot be the correct solution because the transition between the upper end of the pillar and the upper protruding beam is shown as a single plane According to the view from above however the vertical edge of the uppermost end of the pillar would have to be visible here Question type 2 Example 1 In the case of this simple example you can immediately rule out the perspective from below and above From below as well as from above you would be looking through a kind of tube Therefore the perspective illustrated on the right can only be the view from the right the left or behind Now look at the bottom end of the cable In the left-hand picture it faces you In the right-hand picture it faces away from you ie points in exactly the opposite direction Therefore it is clear that the right-hand picture shows the view from behind
Example 2 Here the only view you can rule out immediately is the one from behind (Option E) If the view from the front shows one end of the cable leading toward the back of the cube at the top right the view from behind would show this cable end coming at you at the top left This is not the case in the right-hand picture If you tip the cube forward in your imagination you immediately see that the correct answer cannot be the view from above and turning the cube 180 degrees or 90 degrees to the right also does not lead to the desired perspective But if you imagine yourself standing on the right side of the cube you see that the end of the cable which is concealed in the left-hand illustration comes toward you on the right side of the cube in the right-hand illustration Therefore right (A) is the correct answer
Example 3 Here the figure on the right cannot be showing the view of the cable from the left (B) In the view from the left the section of the cable running horizontally in the view from the front would have to be visible in the middle of the right-hand edge Answer (E) is incorrect for the same reason In the view from behind the horizontal section of cable would have to be
visible in the background likewise running horizontally about halfway between bottom and top
In the view from above (D) this same section of cable would have to be seen leading from one side to the other along the bottom surface of the cube The figure on the right cannot be showing the view from the right (A) because the part of the cable touching the upper left-hand wall in that figure would have to be touching the upper right-hand wall in the view from the front which is not the case The only remaining option is the view from below (C) which only reveals itself to be correct however upon closer examination We might easily find ourselves looking for the end of the cable clearly seen at the bottom left in the view from the front ndash In the view from below it runs right into a curve in the cable and thus appears not to be an end at all On the other hand the end of the cable visible on the right-hand edge in the view from below is not visible in the view from the front because it is hidden behind a curving section of cable
In the subtest Analysing Technical Interrelationships you have to analyse and interpret diagrams charts or tables depicting technical laws or formulae
The test measures the ability to abstract from scientific and technical facts and to put abstract facts in concrete terms Knowledge of mathematics physics or technology is not needed background information will be provided if necessary
Instructions Working time 60 minutes
These items contain questions from various technical areas Your task is to visualize simple technical procedures and recognize technical interrelationships Unless otherwise indicated the axes (scales) of all diagrams are linearly subdivided In some of the items you must identify the ldquoqualitativelyrdquo correct diagram In other words your task is to decide which graph best represents the relationship between the variables shown Even the correct diagram will not necessarily drawn to scale
Example 1 Arrangements I and II each include a beam which is pivot-mounted (like a swing or see-saw) A hook has been mounted on the right end of the beam The ends of the beam are connected by a rope which is threaded through rolls
beam beam
bearing hook
rope rope
hookbearing
A weight is hung on the hook Which of the following statements isare then correct (The masses of the beam rope and hook can be neglected)
I In the case of Arrangement I the right end of the beam moves downward II In the case of Arrangement II the right end of the beam moves downward
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty low
Example 2 This diagram shows the wattage required by a lift motor in a period of 12 minutes (min) When the lift travels upward four times as much wattage (per minute) is required as when the lift travels downward When the lift stops at a floor twice as much wattage (per minute) is required as when the lift travels downward The running time between two consecutive floors is 30 seconds At the point in time t = 0 the lift is on the third floor
Which of the following statements isare correct
I Within the 12 minutes shown the lift travels up to the 6th floor II At the point in time t = 10 min the lift is on the 3rd floor
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty medium Example 3 The diagram shows two thermometers on which no temperature scales have yet been indicated They are both filled with the same liquid and the amount of liquid is also the same Their tubes are of the same length However the tube of the left-hand thermometer has a narrower diameter than that of the right-hand thermometer
We will assume that the markings for the two temperature scales are added correctly they begin at the same height on each tube and end at the same height Both thermometers are used only at temperatures for which they are suitable
Which of the two statements is (or are) therefore correct
I Rises in temperature can be measured less accurately with the left-hand thermometer than with the right-hand one II The right-hand thermometer covers a greater temperature range than the left-hand one
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty high
Solutions Subtest ldquoAnalysing Technical Interrelationshipsrdquo Example 1 If there is no rope the right end of the beam moves downward in both cases when a weight is suspended from the hook So the question is whether this movement is prevented by the rope When a weight is suspended from the hook in Arrangement I the rope slackens at the right end and at the left end The right end of the beam moves downward and Statement I is accordingly correct When a weight is suspended from the hook in Arrangement II traction (pulling power) is applied to the right end of the rope By way of the rope this traction is transferred to the left end of the beam Since both ends of the beam are pulled downward with the same force the beam does not move Statement II is false The solution to this item is therefore A
Example 2 Three different values for the wattage P are shown in the diagram 1 2 and 4 According to the text P is the lowest when the lift travels downward In this case therefore P = 1 When the lift stops at a floor P = 2 When the lift travels upward P = 4
On the basis of this information it is possible to interpret the movement of the lift At the point in time t = 0 the lift is on the 3rd floor and stops there for 1 minute Then it travels downward for 1 minute Since it takes 30 seconds to travel one floor it is then on the 1st floor Following a stop of 1 minute it travels upward 2 minutes (corresponding to 4 floors) At the point in time t = 5 it is therefore on the 5th floor There it stops for 15 minutes then travelling upward to the 6th floor ndash Statement I is therefore correct One minute later it travels 15 minutes (corresponding to 3 floors) downward and at the point in time t = 95 is therefore on the 3rd floor ndash Statement II is accordingly also correct
The solution to this item is therefore C
Example 3 If the temperature is increased by x degrees the liquid inside each thermometer expands by the same volume However this increase in liquid volume makes the liquid rise to a higher level in the tube of the thermometer on the left Since the cross-section of the tube in the left-hand thermometer is smaller than that of the right-hand one a defined temperature change generally leads here to a greater change in the liquid level than with the right-hand
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
The solution to this item is therefore B
Question type 2
The following items also test your ability to visualise three-dimensional figures Each item consists of two illustrations showing a transparent cube with one or two cables in its interior The first illustration (left) always shows the view from the front In the picture on the right the same cube is illustrated again Your task is to determine whether the picture on the right shows that cube from the right (r) left (l) from below (w) above (a) or behind (d)
(A) r (B) l (C) w (D) a (E) d
Here you see the cube from the front
Here you see the cube from
In the picture on the right you see the cube from above On your answer sheet you would mark the (D)
These items can be solved in one of the following two ways
Imagine that the cube had been placed on a glass table and that you could walk all the way around it Standing to the right or left of the table you look at the cube from the right or from the left If you go behind the table you look at the cube from behind If you come back to the front of the table and bend over it to look at the cube from above and if you imagine yourself lying down underneath the table feet first you see the view from below
Or you imagine that you could pick up the cube and turn it around in your hands If you looked at the cube from the front ie from the position shown in the left-hand illustration and then tipped it towards you by 90 degrees not changing your own position at all then you would see the view from above If you looked at the cube from the front and then turned it 90 degrees to the right you would see the view from the left If you turned it from the starting position 90 degrees to the left you would see it from the right And if you turned it 180 degrees to the right or left from the starting position you would see it from behind Finally if you tipped it backward you would see it from below
Example 1
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty low Example 2
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty medium
Example 3
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front
Here you see the cube from
Degree of difficulty high
Solutions Subtest rdquoVisualising Solidsldquo Question type 1 Example 1 Visualise this solid as a tree stump which has been cut off diagonally When you look at it from above (view from above) you see that a fairly large piece has been cut out of its left half Behind the cut-out section however a relatively large section of the stump has remained standing You can therefore rule out Option (A) immediately because this option shows nothing remaining behind the cut-out section except for the outer bark Option (B) can likewise not be the correct solution since here only a section of bark has been removed from the front The piece cut out of Option (D) has only one straight side According to the view from above however the cut-out section has to have three straight sides This is the case only with the section cut out of Option (C) Therefore (C) is the correct answer Example 2 One means of solving this item is to begin by looking at a detail which does not occur in all four answer options Look for example at the figure which looks like an upside-down L on the left edge of Option (B) and (D) Does this figure result from the view from above and the view from the front Yes it does because the upside-down L is the unobstructed view of the high surface at the figurersquos centre Accordingly you can already rule out Options (A) and (C) Options (B) und (D) differ in that in Figure (B) a step has been indicated over the rectangular figure to the right whereas (D) shows a straight edge all the way to the top In the view from above however you can see the protruding section which forms the step Therefore (B) is the correct answer here
Example 3 In the process of solving this item it is initially unclear whether the side view we are looking for is a view from the left or one from the right On the basis of the view from above it can be deduced that in the view from the left side (VSL) one of the two protruding beams at the upper end of the figure points towards the viewer the other towards the right In the view from the right side (VSR) only one of the two protruding beams can be seen it points towards the left The answers (B) and (D) are therefore options for the VSL and the answers (A) and (C) are options for the VSR Answer (B) is out of the question because there is no line indicating an edge at the transition from the figurersquos base to the vertical element or ldquopillarrdquo on top of it ndash On the basis of the view from above it can be deduced that the base of the figure is square Therefore any view from the side would have to indicate the upper edge of the base just as the view from the front does In the case of answer (D) there is a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam In the view from above however this line cannot be accounted for In the case of answer (A) on the other hand there is likewise a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam Here that line is correct because this is one of the two options for a VSR The left vertical edge of the pillar must therefore be visible in front of the protruding beam The other elements of answer (A) also correspond with the view of the three-dimensional figure from above and the view of it from the side and (A) is accordingly the correct solution to this task Answer (C) cannot be the correct solution because the transition between the upper end of the pillar and the upper protruding beam is shown as a single plane According to the view from above however the vertical edge of the uppermost end of the pillar would have to be visible here Question type 2 Example 1 In the case of this simple example you can immediately rule out the perspective from below and above From below as well as from above you would be looking through a kind of tube Therefore the perspective illustrated on the right can only be the view from the right the left or behind Now look at the bottom end of the cable In the left-hand picture it faces you In the right-hand picture it faces away from you ie points in exactly the opposite direction Therefore it is clear that the right-hand picture shows the view from behind
Example 2 Here the only view you can rule out immediately is the one from behind (Option E) If the view from the front shows one end of the cable leading toward the back of the cube at the top right the view from behind would show this cable end coming at you at the top left This is not the case in the right-hand picture If you tip the cube forward in your imagination you immediately see that the correct answer cannot be the view from above and turning the cube 180 degrees or 90 degrees to the right also does not lead to the desired perspective But if you imagine yourself standing on the right side of the cube you see that the end of the cable which is concealed in the left-hand illustration comes toward you on the right side of the cube in the right-hand illustration Therefore right (A) is the correct answer
Example 3 Here the figure on the right cannot be showing the view of the cable from the left (B) In the view from the left the section of the cable running horizontally in the view from the front would have to be visible in the middle of the right-hand edge Answer (E) is incorrect for the same reason In the view from behind the horizontal section of cable would have to be
visible in the background likewise running horizontally about halfway between bottom and top
In the view from above (D) this same section of cable would have to be seen leading from one side to the other along the bottom surface of the cube The figure on the right cannot be showing the view from the right (A) because the part of the cable touching the upper left-hand wall in that figure would have to be touching the upper right-hand wall in the view from the front which is not the case The only remaining option is the view from below (C) which only reveals itself to be correct however upon closer examination We might easily find ourselves looking for the end of the cable clearly seen at the bottom left in the view from the front ndash In the view from below it runs right into a curve in the cable and thus appears not to be an end at all On the other hand the end of the cable visible on the right-hand edge in the view from below is not visible in the view from the front because it is hidden behind a curving section of cable
In the subtest Analysing Technical Interrelationships you have to analyse and interpret diagrams charts or tables depicting technical laws or formulae
The test measures the ability to abstract from scientific and technical facts and to put abstract facts in concrete terms Knowledge of mathematics physics or technology is not needed background information will be provided if necessary
Instructions Working time 60 minutes
These items contain questions from various technical areas Your task is to visualize simple technical procedures and recognize technical interrelationships Unless otherwise indicated the axes (scales) of all diagrams are linearly subdivided In some of the items you must identify the ldquoqualitativelyrdquo correct diagram In other words your task is to decide which graph best represents the relationship between the variables shown Even the correct diagram will not necessarily drawn to scale
Example 1 Arrangements I and II each include a beam which is pivot-mounted (like a swing or see-saw) A hook has been mounted on the right end of the beam The ends of the beam are connected by a rope which is threaded through rolls
beam beam
bearing hook
rope rope
hookbearing
A weight is hung on the hook Which of the following statements isare then correct (The masses of the beam rope and hook can be neglected)
I In the case of Arrangement I the right end of the beam moves downward II In the case of Arrangement II the right end of the beam moves downward
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty low
Example 2 This diagram shows the wattage required by a lift motor in a period of 12 minutes (min) When the lift travels upward four times as much wattage (per minute) is required as when the lift travels downward When the lift stops at a floor twice as much wattage (per minute) is required as when the lift travels downward The running time between two consecutive floors is 30 seconds At the point in time t = 0 the lift is on the third floor
Which of the following statements isare correct
I Within the 12 minutes shown the lift travels up to the 6th floor II At the point in time t = 10 min the lift is on the 3rd floor
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty medium Example 3 The diagram shows two thermometers on which no temperature scales have yet been indicated They are both filled with the same liquid and the amount of liquid is also the same Their tubes are of the same length However the tube of the left-hand thermometer has a narrower diameter than that of the right-hand thermometer
We will assume that the markings for the two temperature scales are added correctly they begin at the same height on each tube and end at the same height Both thermometers are used only at temperatures for which they are suitable
Which of the two statements is (or are) therefore correct
I Rises in temperature can be measured less accurately with the left-hand thermometer than with the right-hand one II The right-hand thermometer covers a greater temperature range than the left-hand one
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty high
Solutions Subtest ldquoAnalysing Technical Interrelationshipsrdquo Example 1 If there is no rope the right end of the beam moves downward in both cases when a weight is suspended from the hook So the question is whether this movement is prevented by the rope When a weight is suspended from the hook in Arrangement I the rope slackens at the right end and at the left end The right end of the beam moves downward and Statement I is accordingly correct When a weight is suspended from the hook in Arrangement II traction (pulling power) is applied to the right end of the rope By way of the rope this traction is transferred to the left end of the beam Since both ends of the beam are pulled downward with the same force the beam does not move Statement II is false The solution to this item is therefore A
Example 2 Three different values for the wattage P are shown in the diagram 1 2 and 4 According to the text P is the lowest when the lift travels downward In this case therefore P = 1 When the lift stops at a floor P = 2 When the lift travels upward P = 4
On the basis of this information it is possible to interpret the movement of the lift At the point in time t = 0 the lift is on the 3rd floor and stops there for 1 minute Then it travels downward for 1 minute Since it takes 30 seconds to travel one floor it is then on the 1st floor Following a stop of 1 minute it travels upward 2 minutes (corresponding to 4 floors) At the point in time t = 5 it is therefore on the 5th floor There it stops for 15 minutes then travelling upward to the 6th floor ndash Statement I is therefore correct One minute later it travels 15 minutes (corresponding to 3 floors) downward and at the point in time t = 95 is therefore on the 3rd floor ndash Statement II is accordingly also correct
The solution to this item is therefore C
Example 3 If the temperature is increased by x degrees the liquid inside each thermometer expands by the same volume However this increase in liquid volume makes the liquid rise to a higher level in the tube of the thermometer on the left Since the cross-section of the tube in the left-hand thermometer is smaller than that of the right-hand one a defined temperature change generally leads here to a greater change in the liquid level than with the right-hand
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
The solution to this item is therefore B
Example 1
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty low Example 2
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front Here you see the cube from Degree of difficulty medium
Example 3
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front
Here you see the cube from
Degree of difficulty high
Solutions Subtest rdquoVisualising Solidsldquo Question type 1 Example 1 Visualise this solid as a tree stump which has been cut off diagonally When you look at it from above (view from above) you see that a fairly large piece has been cut out of its left half Behind the cut-out section however a relatively large section of the stump has remained standing You can therefore rule out Option (A) immediately because this option shows nothing remaining behind the cut-out section except for the outer bark Option (B) can likewise not be the correct solution since here only a section of bark has been removed from the front The piece cut out of Option (D) has only one straight side According to the view from above however the cut-out section has to have three straight sides This is the case only with the section cut out of Option (C) Therefore (C) is the correct answer Example 2 One means of solving this item is to begin by looking at a detail which does not occur in all four answer options Look for example at the figure which looks like an upside-down L on the left edge of Option (B) and (D) Does this figure result from the view from above and the view from the front Yes it does because the upside-down L is the unobstructed view of the high surface at the figurersquos centre Accordingly you can already rule out Options (A) and (C) Options (B) und (D) differ in that in Figure (B) a step has been indicated over the rectangular figure to the right whereas (D) shows a straight edge all the way to the top In the view from above however you can see the protruding section which forms the step Therefore (B) is the correct answer here
Example 3 In the process of solving this item it is initially unclear whether the side view we are looking for is a view from the left or one from the right On the basis of the view from above it can be deduced that in the view from the left side (VSL) one of the two protruding beams at the upper end of the figure points towards the viewer the other towards the right In the view from the right side (VSR) only one of the two protruding beams can be seen it points towards the left The answers (B) and (D) are therefore options for the VSL and the answers (A) and (C) are options for the VSR Answer (B) is out of the question because there is no line indicating an edge at the transition from the figurersquos base to the vertical element or ldquopillarrdquo on top of it ndash On the basis of the view from above it can be deduced that the base of the figure is square Therefore any view from the side would have to indicate the upper edge of the base just as the view from the front does In the case of answer (D) there is a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam In the view from above however this line cannot be accounted for In the case of answer (A) on the other hand there is likewise a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam Here that line is correct because this is one of the two options for a VSR The left vertical edge of the pillar must therefore be visible in front of the protruding beam The other elements of answer (A) also correspond with the view of the three-dimensional figure from above and the view of it from the side and (A) is accordingly the correct solution to this task Answer (C) cannot be the correct solution because the transition between the upper end of the pillar and the upper protruding beam is shown as a single plane According to the view from above however the vertical edge of the uppermost end of the pillar would have to be visible here Question type 2 Example 1 In the case of this simple example you can immediately rule out the perspective from below and above From below as well as from above you would be looking through a kind of tube Therefore the perspective illustrated on the right can only be the view from the right the left or behind Now look at the bottom end of the cable In the left-hand picture it faces you In the right-hand picture it faces away from you ie points in exactly the opposite direction Therefore it is clear that the right-hand picture shows the view from behind
Example 2 Here the only view you can rule out immediately is the one from behind (Option E) If the view from the front shows one end of the cable leading toward the back of the cube at the top right the view from behind would show this cable end coming at you at the top left This is not the case in the right-hand picture If you tip the cube forward in your imagination you immediately see that the correct answer cannot be the view from above and turning the cube 180 degrees or 90 degrees to the right also does not lead to the desired perspective But if you imagine yourself standing on the right side of the cube you see that the end of the cable which is concealed in the left-hand illustration comes toward you on the right side of the cube in the right-hand illustration Therefore right (A) is the correct answer
Example 3 Here the figure on the right cannot be showing the view of the cable from the left (B) In the view from the left the section of the cable running horizontally in the view from the front would have to be visible in the middle of the right-hand edge Answer (E) is incorrect for the same reason In the view from behind the horizontal section of cable would have to be
visible in the background likewise running horizontally about halfway between bottom and top
In the view from above (D) this same section of cable would have to be seen leading from one side to the other along the bottom surface of the cube The figure on the right cannot be showing the view from the right (A) because the part of the cable touching the upper left-hand wall in that figure would have to be touching the upper right-hand wall in the view from the front which is not the case The only remaining option is the view from below (C) which only reveals itself to be correct however upon closer examination We might easily find ourselves looking for the end of the cable clearly seen at the bottom left in the view from the front ndash In the view from below it runs right into a curve in the cable and thus appears not to be an end at all On the other hand the end of the cable visible on the right-hand edge in the view from below is not visible in the view from the front because it is hidden behind a curving section of cable
In the subtest Analysing Technical Interrelationships you have to analyse and interpret diagrams charts or tables depicting technical laws or formulae
The test measures the ability to abstract from scientific and technical facts and to put abstract facts in concrete terms Knowledge of mathematics physics or technology is not needed background information will be provided if necessary
Instructions Working time 60 minutes
These items contain questions from various technical areas Your task is to visualize simple technical procedures and recognize technical interrelationships Unless otherwise indicated the axes (scales) of all diagrams are linearly subdivided In some of the items you must identify the ldquoqualitativelyrdquo correct diagram In other words your task is to decide which graph best represents the relationship between the variables shown Even the correct diagram will not necessarily drawn to scale
Example 1 Arrangements I and II each include a beam which is pivot-mounted (like a swing or see-saw) A hook has been mounted on the right end of the beam The ends of the beam are connected by a rope which is threaded through rolls
beam beam
bearing hook
rope rope
hookbearing
A weight is hung on the hook Which of the following statements isare then correct (The masses of the beam rope and hook can be neglected)
I In the case of Arrangement I the right end of the beam moves downward II In the case of Arrangement II the right end of the beam moves downward
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty low
Example 2 This diagram shows the wattage required by a lift motor in a period of 12 minutes (min) When the lift travels upward four times as much wattage (per minute) is required as when the lift travels downward When the lift stops at a floor twice as much wattage (per minute) is required as when the lift travels downward The running time between two consecutive floors is 30 seconds At the point in time t = 0 the lift is on the third floor
Which of the following statements isare correct
I Within the 12 minutes shown the lift travels up to the 6th floor II At the point in time t = 10 min the lift is on the 3rd floor
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty medium Example 3 The diagram shows two thermometers on which no temperature scales have yet been indicated They are both filled with the same liquid and the amount of liquid is also the same Their tubes are of the same length However the tube of the left-hand thermometer has a narrower diameter than that of the right-hand thermometer
We will assume that the markings for the two temperature scales are added correctly they begin at the same height on each tube and end at the same height Both thermometers are used only at temperatures for which they are suitable
Which of the two statements is (or are) therefore correct
I Rises in temperature can be measured less accurately with the left-hand thermometer than with the right-hand one II The right-hand thermometer covers a greater temperature range than the left-hand one
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty high
Solutions Subtest ldquoAnalysing Technical Interrelationshipsrdquo Example 1 If there is no rope the right end of the beam moves downward in both cases when a weight is suspended from the hook So the question is whether this movement is prevented by the rope When a weight is suspended from the hook in Arrangement I the rope slackens at the right end and at the left end The right end of the beam moves downward and Statement I is accordingly correct When a weight is suspended from the hook in Arrangement II traction (pulling power) is applied to the right end of the rope By way of the rope this traction is transferred to the left end of the beam Since both ends of the beam are pulled downward with the same force the beam does not move Statement II is false The solution to this item is therefore A
Example 2 Three different values for the wattage P are shown in the diagram 1 2 and 4 According to the text P is the lowest when the lift travels downward In this case therefore P = 1 When the lift stops at a floor P = 2 When the lift travels upward P = 4
On the basis of this information it is possible to interpret the movement of the lift At the point in time t = 0 the lift is on the 3rd floor and stops there for 1 minute Then it travels downward for 1 minute Since it takes 30 seconds to travel one floor it is then on the 1st floor Following a stop of 1 minute it travels upward 2 minutes (corresponding to 4 floors) At the point in time t = 5 it is therefore on the 5th floor There it stops for 15 minutes then travelling upward to the 6th floor ndash Statement I is therefore correct One minute later it travels 15 minutes (corresponding to 3 floors) downward and at the point in time t = 95 is therefore on the 3rd floor ndash Statement II is accordingly also correct
The solution to this item is therefore C
Example 3 If the temperature is increased by x degrees the liquid inside each thermometer expands by the same volume However this increase in liquid volume makes the liquid rise to a higher level in the tube of the thermometer on the left Since the cross-section of the tube in the left-hand thermometer is smaller than that of the right-hand one a defined temperature change generally leads here to a greater change in the liquid level than with the right-hand
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
The solution to this item is therefore B
Example 3
(A) r
(B) l
(C) w
(D) a
(E) d
Here you see the cube from the front
Here you see the cube from
Degree of difficulty high
Solutions Subtest rdquoVisualising Solidsldquo Question type 1 Example 1 Visualise this solid as a tree stump which has been cut off diagonally When you look at it from above (view from above) you see that a fairly large piece has been cut out of its left half Behind the cut-out section however a relatively large section of the stump has remained standing You can therefore rule out Option (A) immediately because this option shows nothing remaining behind the cut-out section except for the outer bark Option (B) can likewise not be the correct solution since here only a section of bark has been removed from the front The piece cut out of Option (D) has only one straight side According to the view from above however the cut-out section has to have three straight sides This is the case only with the section cut out of Option (C) Therefore (C) is the correct answer Example 2 One means of solving this item is to begin by looking at a detail which does not occur in all four answer options Look for example at the figure which looks like an upside-down L on the left edge of Option (B) and (D) Does this figure result from the view from above and the view from the front Yes it does because the upside-down L is the unobstructed view of the high surface at the figurersquos centre Accordingly you can already rule out Options (A) and (C) Options (B) und (D) differ in that in Figure (B) a step has been indicated over the rectangular figure to the right whereas (D) shows a straight edge all the way to the top In the view from above however you can see the protruding section which forms the step Therefore (B) is the correct answer here
Example 3 In the process of solving this item it is initially unclear whether the side view we are looking for is a view from the left or one from the right On the basis of the view from above it can be deduced that in the view from the left side (VSL) one of the two protruding beams at the upper end of the figure points towards the viewer the other towards the right In the view from the right side (VSR) only one of the two protruding beams can be seen it points towards the left The answers (B) and (D) are therefore options for the VSL and the answers (A) and (C) are options for the VSR Answer (B) is out of the question because there is no line indicating an edge at the transition from the figurersquos base to the vertical element or ldquopillarrdquo on top of it ndash On the basis of the view from above it can be deduced that the base of the figure is square Therefore any view from the side would have to indicate the upper edge of the base just as the view from the front does In the case of answer (D) there is a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam In the view from above however this line cannot be accounted for In the case of answer (A) on the other hand there is likewise a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam Here that line is correct because this is one of the two options for a VSR The left vertical edge of the pillar must therefore be visible in front of the protruding beam The other elements of answer (A) also correspond with the view of the three-dimensional figure from above and the view of it from the side and (A) is accordingly the correct solution to this task Answer (C) cannot be the correct solution because the transition between the upper end of the pillar and the upper protruding beam is shown as a single plane According to the view from above however the vertical edge of the uppermost end of the pillar would have to be visible here Question type 2 Example 1 In the case of this simple example you can immediately rule out the perspective from below and above From below as well as from above you would be looking through a kind of tube Therefore the perspective illustrated on the right can only be the view from the right the left or behind Now look at the bottom end of the cable In the left-hand picture it faces you In the right-hand picture it faces away from you ie points in exactly the opposite direction Therefore it is clear that the right-hand picture shows the view from behind
Example 2 Here the only view you can rule out immediately is the one from behind (Option E) If the view from the front shows one end of the cable leading toward the back of the cube at the top right the view from behind would show this cable end coming at you at the top left This is not the case in the right-hand picture If you tip the cube forward in your imagination you immediately see that the correct answer cannot be the view from above and turning the cube 180 degrees or 90 degrees to the right also does not lead to the desired perspective But if you imagine yourself standing on the right side of the cube you see that the end of the cable which is concealed in the left-hand illustration comes toward you on the right side of the cube in the right-hand illustration Therefore right (A) is the correct answer
Example 3 Here the figure on the right cannot be showing the view of the cable from the left (B) In the view from the left the section of the cable running horizontally in the view from the front would have to be visible in the middle of the right-hand edge Answer (E) is incorrect for the same reason In the view from behind the horizontal section of cable would have to be
visible in the background likewise running horizontally about halfway between bottom and top
In the view from above (D) this same section of cable would have to be seen leading from one side to the other along the bottom surface of the cube The figure on the right cannot be showing the view from the right (A) because the part of the cable touching the upper left-hand wall in that figure would have to be touching the upper right-hand wall in the view from the front which is not the case The only remaining option is the view from below (C) which only reveals itself to be correct however upon closer examination We might easily find ourselves looking for the end of the cable clearly seen at the bottom left in the view from the front ndash In the view from below it runs right into a curve in the cable and thus appears not to be an end at all On the other hand the end of the cable visible on the right-hand edge in the view from below is not visible in the view from the front because it is hidden behind a curving section of cable
In the subtest Analysing Technical Interrelationships you have to analyse and interpret diagrams charts or tables depicting technical laws or formulae
The test measures the ability to abstract from scientific and technical facts and to put abstract facts in concrete terms Knowledge of mathematics physics or technology is not needed background information will be provided if necessary
Instructions Working time 60 minutes
These items contain questions from various technical areas Your task is to visualize simple technical procedures and recognize technical interrelationships Unless otherwise indicated the axes (scales) of all diagrams are linearly subdivided In some of the items you must identify the ldquoqualitativelyrdquo correct diagram In other words your task is to decide which graph best represents the relationship between the variables shown Even the correct diagram will not necessarily drawn to scale
Example 1 Arrangements I and II each include a beam which is pivot-mounted (like a swing or see-saw) A hook has been mounted on the right end of the beam The ends of the beam are connected by a rope which is threaded through rolls
beam beam
bearing hook
rope rope
hookbearing
A weight is hung on the hook Which of the following statements isare then correct (The masses of the beam rope and hook can be neglected)
I In the case of Arrangement I the right end of the beam moves downward II In the case of Arrangement II the right end of the beam moves downward
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty low
Example 2 This diagram shows the wattage required by a lift motor in a period of 12 minutes (min) When the lift travels upward four times as much wattage (per minute) is required as when the lift travels downward When the lift stops at a floor twice as much wattage (per minute) is required as when the lift travels downward The running time between two consecutive floors is 30 seconds At the point in time t = 0 the lift is on the third floor
Which of the following statements isare correct
I Within the 12 minutes shown the lift travels up to the 6th floor II At the point in time t = 10 min the lift is on the 3rd floor
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty medium Example 3 The diagram shows two thermometers on which no temperature scales have yet been indicated They are both filled with the same liquid and the amount of liquid is also the same Their tubes are of the same length However the tube of the left-hand thermometer has a narrower diameter than that of the right-hand thermometer
We will assume that the markings for the two temperature scales are added correctly they begin at the same height on each tube and end at the same height Both thermometers are used only at temperatures for which they are suitable
Which of the two statements is (or are) therefore correct
I Rises in temperature can be measured less accurately with the left-hand thermometer than with the right-hand one II The right-hand thermometer covers a greater temperature range than the left-hand one
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty high
Solutions Subtest ldquoAnalysing Technical Interrelationshipsrdquo Example 1 If there is no rope the right end of the beam moves downward in both cases when a weight is suspended from the hook So the question is whether this movement is prevented by the rope When a weight is suspended from the hook in Arrangement I the rope slackens at the right end and at the left end The right end of the beam moves downward and Statement I is accordingly correct When a weight is suspended from the hook in Arrangement II traction (pulling power) is applied to the right end of the rope By way of the rope this traction is transferred to the left end of the beam Since both ends of the beam are pulled downward with the same force the beam does not move Statement II is false The solution to this item is therefore A
Example 2 Three different values for the wattage P are shown in the diagram 1 2 and 4 According to the text P is the lowest when the lift travels downward In this case therefore P = 1 When the lift stops at a floor P = 2 When the lift travels upward P = 4
On the basis of this information it is possible to interpret the movement of the lift At the point in time t = 0 the lift is on the 3rd floor and stops there for 1 minute Then it travels downward for 1 minute Since it takes 30 seconds to travel one floor it is then on the 1st floor Following a stop of 1 minute it travels upward 2 minutes (corresponding to 4 floors) At the point in time t = 5 it is therefore on the 5th floor There it stops for 15 minutes then travelling upward to the 6th floor ndash Statement I is therefore correct One minute later it travels 15 minutes (corresponding to 3 floors) downward and at the point in time t = 95 is therefore on the 3rd floor ndash Statement II is accordingly also correct
The solution to this item is therefore C
Example 3 If the temperature is increased by x degrees the liquid inside each thermometer expands by the same volume However this increase in liquid volume makes the liquid rise to a higher level in the tube of the thermometer on the left Since the cross-section of the tube in the left-hand thermometer is smaller than that of the right-hand one a defined temperature change generally leads here to a greater change in the liquid level than with the right-hand
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
The solution to this item is therefore B
Example 3 In the process of solving this item it is initially unclear whether the side view we are looking for is a view from the left or one from the right On the basis of the view from above it can be deduced that in the view from the left side (VSL) one of the two protruding beams at the upper end of the figure points towards the viewer the other towards the right In the view from the right side (VSR) only one of the two protruding beams can be seen it points towards the left The answers (B) and (D) are therefore options for the VSL and the answers (A) and (C) are options for the VSR Answer (B) is out of the question because there is no line indicating an edge at the transition from the figurersquos base to the vertical element or ldquopillarrdquo on top of it ndash On the basis of the view from above it can be deduced that the base of the figure is square Therefore any view from the side would have to indicate the upper edge of the base just as the view from the front does In the case of answer (D) there is a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam In the view from above however this line cannot be accounted for In the case of answer (A) on the other hand there is likewise a vertical line at the transition between the upper end of the ldquopillarrdquo and the protruding beam Here that line is correct because this is one of the two options for a VSR The left vertical edge of the pillar must therefore be visible in front of the protruding beam The other elements of answer (A) also correspond with the view of the three-dimensional figure from above and the view of it from the side and (A) is accordingly the correct solution to this task Answer (C) cannot be the correct solution because the transition between the upper end of the pillar and the upper protruding beam is shown as a single plane According to the view from above however the vertical edge of the uppermost end of the pillar would have to be visible here Question type 2 Example 1 In the case of this simple example you can immediately rule out the perspective from below and above From below as well as from above you would be looking through a kind of tube Therefore the perspective illustrated on the right can only be the view from the right the left or behind Now look at the bottom end of the cable In the left-hand picture it faces you In the right-hand picture it faces away from you ie points in exactly the opposite direction Therefore it is clear that the right-hand picture shows the view from behind
Example 2 Here the only view you can rule out immediately is the one from behind (Option E) If the view from the front shows one end of the cable leading toward the back of the cube at the top right the view from behind would show this cable end coming at you at the top left This is not the case in the right-hand picture If you tip the cube forward in your imagination you immediately see that the correct answer cannot be the view from above and turning the cube 180 degrees or 90 degrees to the right also does not lead to the desired perspective But if you imagine yourself standing on the right side of the cube you see that the end of the cable which is concealed in the left-hand illustration comes toward you on the right side of the cube in the right-hand illustration Therefore right (A) is the correct answer
Example 3 Here the figure on the right cannot be showing the view of the cable from the left (B) In the view from the left the section of the cable running horizontally in the view from the front would have to be visible in the middle of the right-hand edge Answer (E) is incorrect for the same reason In the view from behind the horizontal section of cable would have to be
visible in the background likewise running horizontally about halfway between bottom and top
In the view from above (D) this same section of cable would have to be seen leading from one side to the other along the bottom surface of the cube The figure on the right cannot be showing the view from the right (A) because the part of the cable touching the upper left-hand wall in that figure would have to be touching the upper right-hand wall in the view from the front which is not the case The only remaining option is the view from below (C) which only reveals itself to be correct however upon closer examination We might easily find ourselves looking for the end of the cable clearly seen at the bottom left in the view from the front ndash In the view from below it runs right into a curve in the cable and thus appears not to be an end at all On the other hand the end of the cable visible on the right-hand edge in the view from below is not visible in the view from the front because it is hidden behind a curving section of cable
In the subtest Analysing Technical Interrelationships you have to analyse and interpret diagrams charts or tables depicting technical laws or formulae
The test measures the ability to abstract from scientific and technical facts and to put abstract facts in concrete terms Knowledge of mathematics physics or technology is not needed background information will be provided if necessary
Instructions Working time 60 minutes
These items contain questions from various technical areas Your task is to visualize simple technical procedures and recognize technical interrelationships Unless otherwise indicated the axes (scales) of all diagrams are linearly subdivided In some of the items you must identify the ldquoqualitativelyrdquo correct diagram In other words your task is to decide which graph best represents the relationship between the variables shown Even the correct diagram will not necessarily drawn to scale
Example 1 Arrangements I and II each include a beam which is pivot-mounted (like a swing or see-saw) A hook has been mounted on the right end of the beam The ends of the beam are connected by a rope which is threaded through rolls
beam beam
bearing hook
rope rope
hookbearing
A weight is hung on the hook Which of the following statements isare then correct (The masses of the beam rope and hook can be neglected)
I In the case of Arrangement I the right end of the beam moves downward II In the case of Arrangement II the right end of the beam moves downward
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty low
Example 2 This diagram shows the wattage required by a lift motor in a period of 12 minutes (min) When the lift travels upward four times as much wattage (per minute) is required as when the lift travels downward When the lift stops at a floor twice as much wattage (per minute) is required as when the lift travels downward The running time between two consecutive floors is 30 seconds At the point in time t = 0 the lift is on the third floor
Which of the following statements isare correct
I Within the 12 minutes shown the lift travels up to the 6th floor II At the point in time t = 10 min the lift is on the 3rd floor
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty medium Example 3 The diagram shows two thermometers on which no temperature scales have yet been indicated They are both filled with the same liquid and the amount of liquid is also the same Their tubes are of the same length However the tube of the left-hand thermometer has a narrower diameter than that of the right-hand thermometer
We will assume that the markings for the two temperature scales are added correctly they begin at the same height on each tube and end at the same height Both thermometers are used only at temperatures for which they are suitable
Which of the two statements is (or are) therefore correct
I Rises in temperature can be measured less accurately with the left-hand thermometer than with the right-hand one II The right-hand thermometer covers a greater temperature range than the left-hand one
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty high
Solutions Subtest ldquoAnalysing Technical Interrelationshipsrdquo Example 1 If there is no rope the right end of the beam moves downward in both cases when a weight is suspended from the hook So the question is whether this movement is prevented by the rope When a weight is suspended from the hook in Arrangement I the rope slackens at the right end and at the left end The right end of the beam moves downward and Statement I is accordingly correct When a weight is suspended from the hook in Arrangement II traction (pulling power) is applied to the right end of the rope By way of the rope this traction is transferred to the left end of the beam Since both ends of the beam are pulled downward with the same force the beam does not move Statement II is false The solution to this item is therefore A
Example 2 Three different values for the wattage P are shown in the diagram 1 2 and 4 According to the text P is the lowest when the lift travels downward In this case therefore P = 1 When the lift stops at a floor P = 2 When the lift travels upward P = 4
On the basis of this information it is possible to interpret the movement of the lift At the point in time t = 0 the lift is on the 3rd floor and stops there for 1 minute Then it travels downward for 1 minute Since it takes 30 seconds to travel one floor it is then on the 1st floor Following a stop of 1 minute it travels upward 2 minutes (corresponding to 4 floors) At the point in time t = 5 it is therefore on the 5th floor There it stops for 15 minutes then travelling upward to the 6th floor ndash Statement I is therefore correct One minute later it travels 15 minutes (corresponding to 3 floors) downward and at the point in time t = 95 is therefore on the 3rd floor ndash Statement II is accordingly also correct
The solution to this item is therefore C
Example 3 If the temperature is increased by x degrees the liquid inside each thermometer expands by the same volume However this increase in liquid volume makes the liquid rise to a higher level in the tube of the thermometer on the left Since the cross-section of the tube in the left-hand thermometer is smaller than that of the right-hand one a defined temperature change generally leads here to a greater change in the liquid level than with the right-hand
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
The solution to this item is therefore B
visible in the background likewise running horizontally about halfway between bottom and top
In the view from above (D) this same section of cable would have to be seen leading from one side to the other along the bottom surface of the cube The figure on the right cannot be showing the view from the right (A) because the part of the cable touching the upper left-hand wall in that figure would have to be touching the upper right-hand wall in the view from the front which is not the case The only remaining option is the view from below (C) which only reveals itself to be correct however upon closer examination We might easily find ourselves looking for the end of the cable clearly seen at the bottom left in the view from the front ndash In the view from below it runs right into a curve in the cable and thus appears not to be an end at all On the other hand the end of the cable visible on the right-hand edge in the view from below is not visible in the view from the front because it is hidden behind a curving section of cable
In the subtest Analysing Technical Interrelationships you have to analyse and interpret diagrams charts or tables depicting technical laws or formulae
The test measures the ability to abstract from scientific and technical facts and to put abstract facts in concrete terms Knowledge of mathematics physics or technology is not needed background information will be provided if necessary
Instructions Working time 60 minutes
These items contain questions from various technical areas Your task is to visualize simple technical procedures and recognize technical interrelationships Unless otherwise indicated the axes (scales) of all diagrams are linearly subdivided In some of the items you must identify the ldquoqualitativelyrdquo correct diagram In other words your task is to decide which graph best represents the relationship between the variables shown Even the correct diagram will not necessarily drawn to scale
Example 1 Arrangements I and II each include a beam which is pivot-mounted (like a swing or see-saw) A hook has been mounted on the right end of the beam The ends of the beam are connected by a rope which is threaded through rolls
beam beam
bearing hook
rope rope
hookbearing
A weight is hung on the hook Which of the following statements isare then correct (The masses of the beam rope and hook can be neglected)
I In the case of Arrangement I the right end of the beam moves downward II In the case of Arrangement II the right end of the beam moves downward
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty low
Example 2 This diagram shows the wattage required by a lift motor in a period of 12 minutes (min) When the lift travels upward four times as much wattage (per minute) is required as when the lift travels downward When the lift stops at a floor twice as much wattage (per minute) is required as when the lift travels downward The running time between two consecutive floors is 30 seconds At the point in time t = 0 the lift is on the third floor
Which of the following statements isare correct
I Within the 12 minutes shown the lift travels up to the 6th floor II At the point in time t = 10 min the lift is on the 3rd floor
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty medium Example 3 The diagram shows two thermometers on which no temperature scales have yet been indicated They are both filled with the same liquid and the amount of liquid is also the same Their tubes are of the same length However the tube of the left-hand thermometer has a narrower diameter than that of the right-hand thermometer
We will assume that the markings for the two temperature scales are added correctly they begin at the same height on each tube and end at the same height Both thermometers are used only at temperatures for which they are suitable
Which of the two statements is (or are) therefore correct
I Rises in temperature can be measured less accurately with the left-hand thermometer than with the right-hand one II The right-hand thermometer covers a greater temperature range than the left-hand one
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty high
Solutions Subtest ldquoAnalysing Technical Interrelationshipsrdquo Example 1 If there is no rope the right end of the beam moves downward in both cases when a weight is suspended from the hook So the question is whether this movement is prevented by the rope When a weight is suspended from the hook in Arrangement I the rope slackens at the right end and at the left end The right end of the beam moves downward and Statement I is accordingly correct When a weight is suspended from the hook in Arrangement II traction (pulling power) is applied to the right end of the rope By way of the rope this traction is transferred to the left end of the beam Since both ends of the beam are pulled downward with the same force the beam does not move Statement II is false The solution to this item is therefore A
Example 2 Three different values for the wattage P are shown in the diagram 1 2 and 4 According to the text P is the lowest when the lift travels downward In this case therefore P = 1 When the lift stops at a floor P = 2 When the lift travels upward P = 4
On the basis of this information it is possible to interpret the movement of the lift At the point in time t = 0 the lift is on the 3rd floor and stops there for 1 minute Then it travels downward for 1 minute Since it takes 30 seconds to travel one floor it is then on the 1st floor Following a stop of 1 minute it travels upward 2 minutes (corresponding to 4 floors) At the point in time t = 5 it is therefore on the 5th floor There it stops for 15 minutes then travelling upward to the 6th floor ndash Statement I is therefore correct One minute later it travels 15 minutes (corresponding to 3 floors) downward and at the point in time t = 95 is therefore on the 3rd floor ndash Statement II is accordingly also correct
The solution to this item is therefore C
Example 3 If the temperature is increased by x degrees the liquid inside each thermometer expands by the same volume However this increase in liquid volume makes the liquid rise to a higher level in the tube of the thermometer on the left Since the cross-section of the tube in the left-hand thermometer is smaller than that of the right-hand one a defined temperature change generally leads here to a greater change in the liquid level than with the right-hand
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
In the subtest Analysing Technical Interrelationships you have to analyse and interpret diagrams charts or tables depicting technical laws or formulae
The test measures the ability to abstract from scientific and technical facts and to put abstract facts in concrete terms Knowledge of mathematics physics or technology is not needed background information will be provided if necessary
Instructions Working time 60 minutes
These items contain questions from various technical areas Your task is to visualize simple technical procedures and recognize technical interrelationships Unless otherwise indicated the axes (scales) of all diagrams are linearly subdivided In some of the items you must identify the ldquoqualitativelyrdquo correct diagram In other words your task is to decide which graph best represents the relationship between the variables shown Even the correct diagram will not necessarily drawn to scale
Example 1 Arrangements I and II each include a beam which is pivot-mounted (like a swing or see-saw) A hook has been mounted on the right end of the beam The ends of the beam are connected by a rope which is threaded through rolls
beam beam
bearing hook
rope rope
hookbearing
A weight is hung on the hook Which of the following statements isare then correct (The masses of the beam rope and hook can be neglected)
I In the case of Arrangement I the right end of the beam moves downward II In the case of Arrangement II the right end of the beam moves downward
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty low
Example 2 This diagram shows the wattage required by a lift motor in a period of 12 minutes (min) When the lift travels upward four times as much wattage (per minute) is required as when the lift travels downward When the lift stops at a floor twice as much wattage (per minute) is required as when the lift travels downward The running time between two consecutive floors is 30 seconds At the point in time t = 0 the lift is on the third floor
Which of the following statements isare correct
I Within the 12 minutes shown the lift travels up to the 6th floor II At the point in time t = 10 min the lift is on the 3rd floor
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty medium Example 3 The diagram shows two thermometers on which no temperature scales have yet been indicated They are both filled with the same liquid and the amount of liquid is also the same Their tubes are of the same length However the tube of the left-hand thermometer has a narrower diameter than that of the right-hand thermometer
We will assume that the markings for the two temperature scales are added correctly they begin at the same height on each tube and end at the same height Both thermometers are used only at temperatures for which they are suitable
Which of the two statements is (or are) therefore correct
I Rises in temperature can be measured less accurately with the left-hand thermometer than with the right-hand one II The right-hand thermometer covers a greater temperature range than the left-hand one
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty high
Solutions Subtest ldquoAnalysing Technical Interrelationshipsrdquo Example 1 If there is no rope the right end of the beam moves downward in both cases when a weight is suspended from the hook So the question is whether this movement is prevented by the rope When a weight is suspended from the hook in Arrangement I the rope slackens at the right end and at the left end The right end of the beam moves downward and Statement I is accordingly correct When a weight is suspended from the hook in Arrangement II traction (pulling power) is applied to the right end of the rope By way of the rope this traction is transferred to the left end of the beam Since both ends of the beam are pulled downward with the same force the beam does not move Statement II is false The solution to this item is therefore A
Example 2 Three different values for the wattage P are shown in the diagram 1 2 and 4 According to the text P is the lowest when the lift travels downward In this case therefore P = 1 When the lift stops at a floor P = 2 When the lift travels upward P = 4
On the basis of this information it is possible to interpret the movement of the lift At the point in time t = 0 the lift is on the 3rd floor and stops there for 1 minute Then it travels downward for 1 minute Since it takes 30 seconds to travel one floor it is then on the 1st floor Following a stop of 1 minute it travels upward 2 minutes (corresponding to 4 floors) At the point in time t = 5 it is therefore on the 5th floor There it stops for 15 minutes then travelling upward to the 6th floor ndash Statement I is therefore correct One minute later it travels 15 minutes (corresponding to 3 floors) downward and at the point in time t = 95 is therefore on the 3rd floor ndash Statement II is accordingly also correct
The solution to this item is therefore C
Example 3 If the temperature is increased by x degrees the liquid inside each thermometer expands by the same volume However this increase in liquid volume makes the liquid rise to a higher level in the tube of the thermometer on the left Since the cross-section of the tube in the left-hand thermometer is smaller than that of the right-hand one a defined temperature change generally leads here to a greater change in the liquid level than with the right-hand
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
The solution to this item is therefore B
Example 2 This diagram shows the wattage required by a lift motor in a period of 12 minutes (min) When the lift travels upward four times as much wattage (per minute) is required as when the lift travels downward When the lift stops at a floor twice as much wattage (per minute) is required as when the lift travels downward The running time between two consecutive floors is 30 seconds At the point in time t = 0 the lift is on the third floor
Which of the following statements isare correct
I Within the 12 minutes shown the lift travels up to the 6th floor II At the point in time t = 10 min the lift is on the 3rd floor
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty medium Example 3 The diagram shows two thermometers on which no temperature scales have yet been indicated They are both filled with the same liquid and the amount of liquid is also the same Their tubes are of the same length However the tube of the left-hand thermometer has a narrower diameter than that of the right-hand thermometer
We will assume that the markings for the two temperature scales are added correctly they begin at the same height on each tube and end at the same height Both thermometers are used only at temperatures for which they are suitable
Which of the two statements is (or are) therefore correct
I Rises in temperature can be measured less accurately with the left-hand thermometer than with the right-hand one II The right-hand thermometer covers a greater temperature range than the left-hand one
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty high
Solutions Subtest ldquoAnalysing Technical Interrelationshipsrdquo Example 1 If there is no rope the right end of the beam moves downward in both cases when a weight is suspended from the hook So the question is whether this movement is prevented by the rope When a weight is suspended from the hook in Arrangement I the rope slackens at the right end and at the left end The right end of the beam moves downward and Statement I is accordingly correct When a weight is suspended from the hook in Arrangement II traction (pulling power) is applied to the right end of the rope By way of the rope this traction is transferred to the left end of the beam Since both ends of the beam are pulled downward with the same force the beam does not move Statement II is false The solution to this item is therefore A
Example 2 Three different values for the wattage P are shown in the diagram 1 2 and 4 According to the text P is the lowest when the lift travels downward In this case therefore P = 1 When the lift stops at a floor P = 2 When the lift travels upward P = 4
On the basis of this information it is possible to interpret the movement of the lift At the point in time t = 0 the lift is on the 3rd floor and stops there for 1 minute Then it travels downward for 1 minute Since it takes 30 seconds to travel one floor it is then on the 1st floor Following a stop of 1 minute it travels upward 2 minutes (corresponding to 4 floors) At the point in time t = 5 it is therefore on the 5th floor There it stops for 15 minutes then travelling upward to the 6th floor ndash Statement I is therefore correct One minute later it travels 15 minutes (corresponding to 3 floors) downward and at the point in time t = 95 is therefore on the 3rd floor ndash Statement II is accordingly also correct
The solution to this item is therefore C
Example 3 If the temperature is increased by x degrees the liquid inside each thermometer expands by the same volume However this increase in liquid volume makes the liquid rise to a higher level in the tube of the thermometer on the left Since the cross-section of the tube in the left-hand thermometer is smaller than that of the right-hand one a defined temperature change generally leads here to a greater change in the liquid level than with the right-hand
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
The solution to this item is therefore B
We will assume that the markings for the two temperature scales are added correctly they begin at the same height on each tube and end at the same height Both thermometers are used only at temperatures for which they are suitable
Which of the two statements is (or are) therefore correct
I Rises in temperature can be measured less accurately with the left-hand thermometer than with the right-hand one II The right-hand thermometer covers a greater temperature range than the left-hand one
(A) Only statement I is correct (B) Only statement II is correct (C) Both statements are correct (D) Neither of the two statements is correct Degree of difficulty high
Solutions Subtest ldquoAnalysing Technical Interrelationshipsrdquo Example 1 If there is no rope the right end of the beam moves downward in both cases when a weight is suspended from the hook So the question is whether this movement is prevented by the rope When a weight is suspended from the hook in Arrangement I the rope slackens at the right end and at the left end The right end of the beam moves downward and Statement I is accordingly correct When a weight is suspended from the hook in Arrangement II traction (pulling power) is applied to the right end of the rope By way of the rope this traction is transferred to the left end of the beam Since both ends of the beam are pulled downward with the same force the beam does not move Statement II is false The solution to this item is therefore A
Example 2 Three different values for the wattage P are shown in the diagram 1 2 and 4 According to the text P is the lowest when the lift travels downward In this case therefore P = 1 When the lift stops at a floor P = 2 When the lift travels upward P = 4
On the basis of this information it is possible to interpret the movement of the lift At the point in time t = 0 the lift is on the 3rd floor and stops there for 1 minute Then it travels downward for 1 minute Since it takes 30 seconds to travel one floor it is then on the 1st floor Following a stop of 1 minute it travels upward 2 minutes (corresponding to 4 floors) At the point in time t = 5 it is therefore on the 5th floor There it stops for 15 minutes then travelling upward to the 6th floor ndash Statement I is therefore correct One minute later it travels 15 minutes (corresponding to 3 floors) downward and at the point in time t = 95 is therefore on the 3rd floor ndash Statement II is accordingly also correct
The solution to this item is therefore C
Example 3 If the temperature is increased by x degrees the liquid inside each thermometer expands by the same volume However this increase in liquid volume makes the liquid rise to a higher level in the tube of the thermometer on the left Since the cross-section of the tube in the left-hand thermometer is smaller than that of the right-hand one a defined temperature change generally leads here to a greater change in the liquid level than with the right-hand
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
The solution to this item is therefore B
thermometer Consequently temperature changes can be measured more accurately with the left-hand thermometer than with the right-hand one Statement I is therefore false
Since a rise in temperature has a lesser effect on the liquid level in the tube of the right-hand thermometer than on that in the left-hand one greater changes in temperature can be measured with the right-hand thermometer The right-hand thermometer thus covers a larger temperature range Statement II is therefore correct
![[southafrica]Formalising South Africa's National Space ... · Microsoft PowerPoint - [southafrica]Formalising South Africa's National Space Programme.PPT Author: wickrama Created](https://static.documents.pub/doc/80x56/6059686d068ebf00f0327038/southafricaformalising-south-africas-national-space-microsoft-powerpoint.jpg)
